Endomorphism algebras of categorified projectors and flag Hilbert scheme charts

Let $P_T$ be the categorified Jones--Wenzl projector indexed by a standard Young tableau $T$ of size $n$, and let $\mathring{\mathrm{FHilb}}_T(\mathbb{C})$ be the corresponding affine chart of the flag Hilbert scheme. Write $\mathcal{A}_T(\mathbb{C})=\Gamma(\mathring{\mathrm{FHilb}}_T(\mathbb{C}),\mathcal{O}_{\mathrm{FHilb}_n^{\mathrm{dg}}(\mathbb{C})})$. The endomorphism-algebra conjecture. The endomorphism algebra of $P_T$ is isomorphic to

$$\operatorname{End}(P_T)=\mathcal{A}_T(\mathbb{C})\otimes\left(\wedge^{\bullet}\mathcal{T}_n^{\vee}|_{\mathring{\mathrm{FHilb}}_T(\mathbb{C})}\right).$$

This proposes a geometric description of the categorified Hecke-algebra idempotents in terms of dg functions on affine flag-Hilbert-scheme charts, together with the exterior factor recording the additional grading.