The $E_n$-enhancement conjecture for homological epimorphisms

Let $F\colon\mathcal A\to\mathcal B$ be a homological epimorphism of $A_\infty$-algebras, and let $\widetilde{\mathcal A}$ be an $E_n$-algebra enhancing $A$. The $E_n$-enhancement conjecture says that there is an $E_n$-algebra $\widetilde{\mathcal B}$ enhancing $\mathcal B$ and a map $\widetilde F\colon\widetilde{\mathcal A}\to\widetilde{\mathcal B}$ enhancing $F$. This conjecture asks whether homological epimorphisms admit compatible $E_n$-algebra enhancements; the source gives no resolution.