Typical periodic optimization for open expanding multi-valued dynamical systems

Let $(X,T)$ be an open expanding multi-valued dynamical system, and let $\operatorname{Lip}(X)$ denote the Banach space of Lipschitz real-valued functions on $X$. Let $P\subseteq\operatorname{Lip}(X)$ consist of those functions whose maximizing measure is unique and supported on a single periodic orbit. TPO conjecture. The set $P$ contains an open dense subset of $\operatorname{Lip}(X)$ and is a prevalent subset of $\operatorname{Lip}(X)$. This extends typical periodic optimization from single-valued systems to the open expanding multi-valued setting; the supplied excerpt gives no resolution status.