Distinguished $\a0\a0\mathbb{P}^n$-structure with prescribed initial data

Let $F\colon D({\mathcal{A}})\to D({\mathcal{B}})$ be an enhanceable functor, and suppose that $(H,Q_n,\gamma)$ is a $\mathbb{P}^n$-structure on $F$. Let $\sigma\colon H[-1]\to {\mathcal{A}}$ make $Q_1$ a coextension of $\operatorname{Id}$ by $H$, and let $\gamma_1\colon Q_1\to RF$ be the specified map. Distinguished-structure conjecture. There is a $\mathbb{P}^n$-structure on $F$ with the same $H$, $\sigma$, and $\gamma_1$, hence producing the same $\mathbb{P}$-twist, such that $Q_n$ is isomorphic, as an object of $D({\mathcal{A}}\text{-}{\mathcal{A}})$, to the truncated twisted tensor algebra. The preceding discussion notes that such a structure need not be unique, while the conjecture asserts the existence of a distinguished one whose cyclic coextension structure is determined by $\sigma$; no resolution is supplied.