Formal Swiss Cheese conjecture for enriched 2-operads

Let $V$ be a symmetric monoidal category and let $\mathcal{P}$ be a $V$-enriched $2$-operad. Let $\mathcal{P}_1$ be the $1$-operad obtained by restricting $\mathcal{P}$ to the $1$-terminal $2$-level trees, and let $\mathrm{Act}^{\mathcal{P}}(A)$ denote the category of actions of $\mathcal{P}$ on a $\mathcal{P}_1$-algebra $A$. Assume that the multitensor associated to $\mathcal{P}$ is closed from one side, with a right-adjoint internal Hom. Let $\mathrm{Sym}_2(\mathcal{P})$ be the symmetrisation of $\mathcal{P}$, and let $\mathrm{Hoch}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(A)=[A,A](\mathrm{id},\mathrm{id})$. Formal Swiss Cheese conjecture. For a $1$-terminal $\mathcal{P}_1$-algebra $A$, the set-theoretical fiber $\mathrm{Act}^{\mathcal{P}}(A)$ is equivalent to the comma category

$$\mathrm{Alg}(\mathrm{Sym}_2(\mathcal{P}))/\mathrm{Hoch}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(A).$$

This gives a strong formal mechanism for describing Swiss Cheese actions through the internal Hochschild object; the supplied text presents it as a conceptual scheme under the stated closedness assumption, without specifying a resolution status.