Kontsevich's tangent-map conjecture for formality quasi-isomorphisms

Let $X$ be a smooth super-manifold, and let $T^\bullet_{\mathrm{poly}}(X)$ and $\mathcal D^\bullet_{\mathrm{poly}}(X)$ denote the differential graded Lie algebras of smooth polyvector fields and smooth polydifferential operators. Let $\mathcal U:T^\bullet_{\mathrm{poly}}(X)\to\mathcal D^\bullet_{\mathrm{poly}}(X)$ be an $L_\infty$-quasi-isomorphism constructed in Section 7 of [K1]. For $\gamma\in T^1_{\mathrm{poly}}(X)$ satisfying

$$[\gamma,\gamma]=0,$$

let $T_\gamma$ denote the tangent complex at $\gamma$ and let $\mathcal U_\gamma$ be the tangent map.

Kontsevich's tangent-map conjecture. The induced map

$$[\mathcal U_\gamma]:H^\bullet T_\gamma(T^\bullet_{\mathrm{poly}}(X))\to H^\bullet T_{\mathcal U(\gamma)}(\mathcal D^\bullet_{\mathrm{poly}}(X))$$

is a morphism of algebras. The conjecture is the formality statement required to obtain a multiplicative Duflo-type map after twisting by a Maurer–Cartan element; its resolution status is not specified in the source.