Formality conjecture for the cyclic bar complex of the Soergel Hecke category

Let $W$ be a Coxeter group with realization $V$ over $\mathbb{C}$, and let $\mathcal{C}(W)$ denote the corresponding Soergel Hecke category. Let $\operatorname{Tr}$ be its universal dg monoidal trace, let $\mathbf{C}_*(\mathcal{C}(W))$ be the cyclic bar complex, and let $\mathrm{HH}_{\bullet}(\mathcal{C}(W))$ be Hochschild homology. Cyclic-bar formality conjecture. The cyclic bar complex

$$\mathbf{C}_*(\mathcal{C}(W))\simeq \operatorname{End}_{\operatorname{Tr}(\mathcal{C}(W))}(\operatorname{Tr}(\mathbb{1}))$$

is formal as a dg algebra. In particular, all maps $\mu_d$ on $\mathrm{HH}_{\bullet}(\mathcal{C}(W))$ vanish for $d\ge 3$. The surrounding argument establishes collapse of a spectral sequence and vanishing of Massey products in a related setting, but the displayed formality assertion is stated as a conjecture.