Converse prisoner-set inclusion conjecture for quadratic mutations

Let $f$ be a quadratic replication system with intact quadratic map $f_{c_1}$, mutation map $f_{c_0}$, and transient region $\operatorname{D}(R)$ of radius $R$. Write $\mathcal{P}(f)$ and $\mathcal{P}(f_{c_1})$ for the corresponding prisoner sets.

Converse prisoner-set inclusion conjecture. If the radius $R$ is such that $\mathcal{P}(f)\subseteq\mathcal{P}(f_{c_1})$, then

$$\operatorname{D}(R)\subseteq\mathcal{P}(f_{c_1}).$$

This is proposed as the converse of the preceding lemma, under general assumptions on the maps and the two radii; the source does not provide a resolution.