The modular-operad isomorphism conjecture for the quantum open operad

Let $\mathcal{QO}$ be the modular operad of quantum open-string vertices, and let $\mathcal{A}\mathit{ss}$ be its genus-zero cyclic suboperad of classical open-string vertices. There are evident morphisms

$$\mathcal{A}\mathit{ss}\longrightarrow\mathcal{QO},\qquad \mathcal{A}\mathit{ss}\longrightarrow\textbf{Mod}\left(\mathcal{A}\mathit{ss}\right).$$

Modular-operad isomorphism conjecture. There is a modular operad isomorphism

$$\mathcal{QO}\cong\textbf{Mod}\left(\mathcal{A}\mathit{ss}\right)$$

compatible with these obvious morphisms. This is the precise algebraic form of the preceding question about whether the quantum open operad is the modular envelope of its genus-zero part. The source cites evidence in favor but does not establish the isomorphism.