The colored full-twist dg-algebra conjecture

For ${\bm{b}}=(b_1,\ldots,b_m)$, let $E_{\bm{b}}$ be the colored polynomial algebra and let $I_{\bm{b}}\vartriangleleft E_{\bm{b}}$ be the colored full-twist ideal. Let $X_{\bm{b}}$ be the blowup of $\operatorname{Spec}(E_{\bm{b}})$ at $I_{\bm{b}}$, with Rees algebra $\mathcal{A}_{\bm{b}}$. Let $\mathcal{B}_{\bm{b}}$ be the colored full-twist dg algebra formed from the morphism complexes of powers of the colored full twist. Colored full-twist dg-algebra conjecture. The natural map of dg algebras

$$\mathcal{B}_{\bm{b}}\longrightarrow\mathcal{A}_{\bm{b}}$$

is a quasi-isomorphism. This proposes a colored analogue of the relationship between Soergel bimodules and the isospectral Hilbert scheme; it is open in general.