The definable $T_5$ spectrum conjecture

Let $M$ be a first-order topological structure, and let $\operatorname{Spec}(A)$ denote the Stone spectrum of a definable set $A\subseteq M^k$. Say that $M$ is definably $T_5$ when every definable subset of every $M^k$ is definably $T_4$. Definable $T_5$ spectrum conjecture. The structure $M$ is definably $T_5$ if and only if, for every definable $A\subseteq M^k$, $\operatorname{Spec}(A)$ is $T_5$. This proposes an equivalence between a definable separation property of the underlying structure and the corresponding separation property of its spectra; the surrounding text gives equivalent formulations and does not state a resolution of this conjecture.