Theorem 1. There is a fully faithful adjoint functor $$ F \colon \mathcal{H}_ \mathrm{mon} \hookrightarrow \mathsf{Ind}(K^b(\mathcal{H}_ \mathrm{mon}^\mathrm{tilt})). $$
Theorem 2. There is a natural action of $\mathcal{H}_ \mathrm{mon}$ on $\mathsf{Shv}_ \mathrm{mon}(\tilde{T})$, which is the same data of a functor $$ \mathbb{V} \colon \mathcal{H}_ \mathrm{mon} \to \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{\tilde{T}}} \times \mathcal{R}_ {I_F^t,\hat{\tilde{T}}}). $$ Then the functor $$ \mathbb{V} \vert_{\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}} \colon \mathcal{H}_ \mathrm{mon}^\mathrm{tilt} \to \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{\tilde{T}}} \times \mathcal{R}_ {I_F^t,\hat{\tilde{T}}})^\heartsuit $$ is fully faithful.
The proof of these statements have some common ingredients. But for now, let us construct this action. Upon choosing an embedding $G \hookrightarrow \GL(V)$, we have a $\tilde{T} \times V_\infty$-torsor (where $V_\infty = I_\infty^{\mathrm{op}+} / I_\infty^{\mathrm{op}++}$) given by $$ \widetilde{\mathrm{Bun}}_ G(\mathbb{P}^1) \to \mathrm{Bun}_ G(\mathbb{P}^1)_ {I_0^+,I_\infty^{\mathrm{op}++}} \to \mathrm{Bun}_ G(\mathbb{P}^1)_ {I_0,I_\infty^{\mathrm{op}+}}. $$ Then we have $\mathcal{H}_ \mathrm{mon}$ acting on $$ \mathsf{Shv}_ \mathrm{mon}(\tilde{T} \backslash \widetilde{\mathrm{Bun}}_ G(\mathbb{P}^1) / (V_\infty, \phi)). $$ Here, this space $\mathrm{Bun}_ G(\mathbb{P}^1)_ {I_0,I_\infty^{\mathrm{op}+}}$ is isomorphic to $\tilde{W}$, but because $\phi$ is nontrivial on each factor, this can only be supported on length zero elements. So writing $$ 1 \to T \to M = \frac{N_{L_\infty(G)}(T) \cap N_{L_\infty(G)}(I^\mathrm{op})}{L_\infty^{\gt 0} T} \to \Omega \to 1, $$ we see that $$ 1 \to Z_G \to M_\phi = Z_M(\phi) \to \Omega \to 1 $$ and we have $$ \mathsf{Shv}_ \mathrm{mon}(\tilde{T} \backslash\backslash \widetilde{\mathrm{Bun}}_ G(\mathbb{P}^1) // (V_\infty, \phi)) \cong \mathsf{Shv}(\tilde{T} \backslash\backslash \tilde{M}) $$ with the right action by $\mathsf{Shv}(\tilde{M}_ \phi)$.
Assume that $Z_G$ is moreover connected. We can then choose a splitting $\Omega \subseteq M_\phi$. Then we can consider the tensor product $$ \mathsf{Shv}(\tilde{T} \backslash\backslash \widetilde{\mathrm{Bun}}_ G(\mathbb{P}^1) // (V_\infty, \phi)) \otimes_{\mathsf{Shv}(\Omega)} \mathsf{Mod}_ \Lambda \cong \mathsf{Shv}(\tilde{T} \backslash\backslash \tilde{T}), $$ where the functor $\mathsf{Shv}(\Omega) \to \mathsf{Mod}_ \Lambda$ is just global sections. This has a natural left action by $\mathcal{H}_ \mathrm{mon}$.
For each $\omega \in \Omega$, we have $$ I_0 \backslash I_0 \omega I_\infty^{\mathrm{op}+} / I_\infty^{\mathrm{op}+} \cong \ast. $$ Let us denote this functor for the neutral component as $$ \Delta_e^\phi \colon \mathsf{Shv}(\tilde{T} \backslash\backslash \tilde{T}) \xrightarrow{\cong} \mathsf{Shv}(\tilde{T} \backslash\backslash \widetilde{\mathrm{Bun}}_ G(\mathbb{P}^1)^0 // (V_\infty,\phi)). $$
Lemma 3. We have $\Delta_{\dot{w}}^\mathrm{mon}(\mathscr{L}) \star^u \Delta_e^\phi(\mathscr{L}^\prime) \cong \Delta_e^\phi(\mathscr{L} \star w(\mathscr{L}^\prime))$ and similarly for $\nabla$.
To prove this, it is enough to show this for $w = s$ a simple reflection. Then this reduces to a computation for $\mathrm{SL}_ 2$ and we can calculate these things on strata.
Corollary 4. We have $$ \mathbb{V}(\Delta_w^\mathrm{mon}(\tilde{\mathrm{Ch}})) \cong \mathbb{V}(\nabla_w^\mathrm{mon}(\tilde{\mathrm{Ch}})) \cong \omega_{\Gamma_w}, $$ where $\Gamma_w$ is the graph of $w$, where $\tilde{W}$ acts on $\hat{\tilde{T}}$.
Recall we have $$ \Hom(\Delta_e^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}), \nabla_w^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}})) = \begin{cases} 0 & e \neq w \br R_\chi & e = w. \end{cases} $$
Lemma 5. We also have $$ \tau^{\le 0} \Hom_{\mathsf{IndCoh}((\mathcal{R}_ {I_F^t,\hat{\tilde{T}}})^2)}(\omega_{\Gamma_e}, \omega_{\Gamma_w}) = \begin{cases} 0 & e \neq w \br R_\chi & e = w. \end{cases} $$
When $w = e$, after truncation this is just the endomorphisms of $\omega$ on $\mathcal{R}_ {I_F^t,\hat{\tilde{T}}}$. When $w \neq e$, this can be thought of as $$ \Hom_{R_\chi \hat{\otimes} R_\chi}(R_\chi, R_\chi(w)), $$ where these are bimodules with the first $R_\chi$ having the natural bimodule structure and the second $R_\chi(w)$ having the natural left action and the $w$-twisted right action. Given any $\varphi \colon R_\chi \to R_\chi(w)$ a bimodule homomorphism, we see that $\varphi(a) = a \varphi(1) = \varphi(1) w(a)$ and hence $(a - w(a)) \varphi(1) = 0$. This implies $\varphi(1) = 0$, because $w$ acted faithfully on $\tilde{T}$.
Corollary 6. We have $$ \Hom(T_{w_1,\chi}^\mathrm{mon}, T_{w_2,\chi}^\mathrm{mon}) \cong \Hom(\mathbb{V}(T_{w_1,\chi}^\mathrm{mon}), \mathbb{V}(T_{w_2,\chi}^\mathrm{mon})). $$
Using the left dual ${}^\vee T_{w_1,\chi}^\mathrm{mon}$, we can identify the left hand side with $$ \Hom(\Delta_e^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}), {}^\vee T_{w_1,\chi}^\mathrm{mon} \star T_{w_2,\chi}^\mathrm{mon}). $$ This dual is also a tilting object and hence the convolution is also a tilting object. Now we can filtered this by costandard objects and apply the lemma.
Recall we had $$ \tilde{W} \supseteq \tilde{W}_ \chi \supseteq \tilde{W}_ \chi^0. $$ Let $r \in \tilde{W}_ \chi^0$ be a simple reflection, which also acts on $\hat{\chi}$. Then we have a closed embedding $$ \hat{\chi} \times_{\hat{\chi} // \lbrace 1, r \rbrace} \hat{\chi} \hookrightarrow \hat{\chi} \times \hat{\chi}. $$
Proposition 7. We have $$ \mathbb{V}(T_{\rho,\chi}^\mathrm{mon}) \cong \omega_{\hat{\chi} \times_{\hat{\chi} // \lbrace 1, r \rbrace} \hat{\chi}}. $$
Here we note that $\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}$ is generated as an idempotent complete additive monoidal category by $T_{\omega,\chi}^\mathrm{mon}$ and $T_{s,\chi}^\mathrm{mon}$, where $\omega \in \Omega$ and $s$ is a simple reflection.
Lemma 8. The category ${}_ \chi \mathcal{H}_ \mathrm{mon,\chi}^\mathrm{tilt}$ is generated as an idempotent complete additive monoidal category by $T_{\omega,\chi}^\mathrm{mon}$ and $T_{r,\chi}^\mathrm{mon}$ where $\omega \in \Omega_\chi$ and $r \in \tilde{W}_ \chi^0$ is a simple reflection.
Definition 9. Fix $\chi$ and define $$ \mathsf{SBim}_ \chi \subseteq \mathsf{IndCoh}(\hat{\chi}^2)^\heartsuit $$ be the idempotent complete additive monoidal category generated by $\omega_{\Gamma_\omega}$ and $\omega_{\hat{\chi} \times_{\hat{\chi} // \lbrace 1, r \rbrace} \hat{\chi}}$ for $\omega \in \Omega_\chi$ and $r \in \tilde{W}_ \chi^0$.
This is now a category that is defined purely on the spectral side.
Theorem 10. The Soelgel bimodule functor defines an equivalence $$ \mathbb{V} \colon {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt} \xrightarrow{\cong} \mathsf{SBim}_ \chi. $$