The lifting conjecture for the singular Hochschild cohomology morphism

Let $A$ be an algebra, let $C_{sg}(A,A)$ denote its singular Hochschild cochain algebra, and let $\operatorname{\mathsf{sg}}_{dg}(A)$ be the dg singularity category of $A$. Write $C(\operatorname{\mathsf{sg}}_{dg}(A),\operatorname{\mathsf{sg}}_{dg}(A))$ for the Hochschild cochain algebra of this dg category. The theorem provides a canonical morphism $\Phi$ from singular Hochschild cohomology to the Hochschild cohomology of $\operatorname{\mathsf{sg}}_{dg}(A)$. Lifting conjecture. The morphism $\Phi$ lifts to a morphism

$$C_{sg}(A,A) \xrightarrow{\sim} C(\operatorname{\mathsf{sg}}_{dg}(A),\operatorname{\mathsf{sg}}_{dg}(A))$$

in the homotopy category of $B_\infty$-algebras. The claim strengthens the theorem's graded-algebra statement by requiring a lift at the cochain level; the supplied text does not indicate whether this lifting assertion has been proved or remains open.