Axiomatization of dynamic topological logic over minimal systems

Let $\mathcal{DTL}^\ast$ be the dynamic topological logic with infinitary temporal modalities, and let $\mathsf L^\ast_{\Diamond f [f]\forall}$ be its monadic language with a universal modality $\forall$. A dynamic topological system is minimal if its underlying space has no non-empty, proper, closed, $f_\mathfrak X$-invariant subset. Let $\mathsf{Ax}_\forall$ denote the standard axioms and rules for the universal modality and its interaction with the other modalities. Minimal-system axiomatization conjecture. The set of formulas in $\mathsf L^\ast_{\Diamond f [f]\forall}$ valid over the class of minimal dynamic topological systems can be axiomatized by

$$\mathcal{DTL}^\ast+\mathsf{Ax}_\forall+\exists\Box p\to\forall\langle f\rangle p.$$

The preceding theorem gives a finite reduction for each formula from validity over minimal systems to validity over all dynamic topological systems. The displayed claim proposes the corresponding uniform axiomatization; the source presents it as future work, and no resolution is supplied.