Betti moduli embedding conjecture for all microsheaves on the affine Springer fiber

Let $B^{\vee,+}$ and $B^{\vee,-}$ be opposite Borel subgroups of $G^{\vee}$ with $B^{\vee,+}\cap B^{\vee,-}=T^{\vee}$. Set $\mathcal{W}=B^{\vee,-}B^{\vee,+}\subset G^{\vee}$, let $\mathcal{W}_0=\mathcal{W}\cap\mathcal{U}^{\vee}$, and let $\widetilde{\mathcal{W}}_0$ be the restriction of the Springer resolution $\widetilde{\mathcal{U}^{\vee}}$ to $\mathcal{W}_0$. The torus $T^{\vee}$ acts by conjugation. Betti moduli embedding conjecture. There is a full embedding

$$\mu\operatorname{Sh}_{\operatorname{Fl}_\psi}(\mathcal{M}_\psi)\hookrightarrow \operatorname{QCoh}^{T^{\vee}}(\widetilde{\mathcal{W}}_0)$$

extending the equivalence in Theorem $D_\psi$--coherent in the stated sense: the Kirillov category is the full subcategory supplied by Theorem mic, while the coherent category is identified with the full subcategory supported on $\mathcal{B}^{\vee}\hookrightarrow\widetilde{\mathcal{W}}_0$, the fiber over $1\in\mathcal{W}_0$. Moreover, this embedding intertwines the natural actions of $\mathbb{X}_*(T)$ on both sides. This would extend the known description of the relevant subcategory of microsheaves to all microsheaves and relate them to coherent sheaves on a Betti moduli space for $G^{\vee}$; the source presents it as a further direction and gives no resolution.