The higher formality conjecture for derived stacks

Let $n\geq 0$, let $X$ be a nice enough derived algebraic stack, and let $\widehat{HH}^{E_{n+1}}(X)$ denote its formal higher Hochschild cohomology, so that $\widehat{HH}^{E_{n+1}}(X)[n+1]$ is a dg-Lie algebra. Let $\mathcal{P}ol(X,n)$ denote the shifted polyvector-field complex of $X$. Higher formality conjecture. The dg-Lie algebra $\widehat{HH}^{E_{n+1}}(X)[n+1]$ is quasi-isomorphic to $\mathcal{P}ol(X,n)[n+1]$, and the quasi-isomorphism is canonical up to a universal choice of a Drinfeld associator. This extends Kontsevich-type formality from ordinary Hochschild cochains to higher Hochschild cohomology of derived algebraic stacks; the statement concerns the existence and associator-dependence of this comparison.