Formality conjecture for the cyclic bar complex of Soergel bimodules

Let $W$ be a Coxeter group with simple reflections $S\subset W$, a realization $V$ over $\mathbb{C}$, and let $\mathrm{SBim}(W)$ be the associated monoidal category of Soergel bimodules. Write $\mathbf{C}(\mathrm{SBim}(W))$ for its cyclic bar complex and $\mathrm{HH}_{\bullet}(\mathrm{SBim}(W))$ for Hochschild homology. Formality conjecture. $\mathbf{C}(\mathrm{SBim}(W))$ is formal as a dg algebra, so higher $A_\infty$-operations on Hochschild homology vanish. The paper proves a closely related formality theorem for the endomorphism algebra of a cube complex, but the stated formality remains a conjecture.