The terminality conjecture for Hochschild Swiss cheese actions

Let $\mathcal{P}$ be a coloured categorical operad and let $A$ be a $\mathcal{P}$-algebra. A Swiss cheese action on $A$ is an action in the sense of the paper, and $\mathrm{Hoch}(A)$ is the Hochschild object associated with $A$.

Terminality conjecture. The object $\mathrm{Hoch}(A)$ is terminal in the homotopy category of Swiss cheese actions on $A$.

This is the universal-property formulation that identifies the Hochschild object as the universal Swiss cheese action. The supplied text does not establish the claim or give resolution evidence, so its status is open.