Local volume-growth conjecture for quasi-optimal pseudomanifolds

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing multiples of a homology class, equipped with the piecewise-flat metric in which every $m$-simplex is isometric to the standard simplex. Let $\varphi_n:\pi_1(P_n)\to\pi_1(X)$ be the homomorphism induced by the representing map, and let $\operatorname{sys}_{\varphi_n}(P_n)$ denote the relative systole. Local volume-growth conjecture. There exist constants $c_m>0$ and $\varepsilon\in(0,\frac12)$ such that, for every $x\in P_n$ and every $r>0$ satisfying

$$\varepsilon\,\operatorname{sys}_{\varphi_n}(P_n)\leqslant r\leqslant\frac12\,\operatorname{sys}_{\varphi_n}(P_n),$$

one has

$$\operatorname{vol} B(x,r)\geqslant c_m r^m.$$

This is suggested by Gromov's property for almost volume-minimizing cycles in systolic geometry, although here the volume being considered is that of pseudomanifolds constructed from the standard simplex. The statement is presented as a possible route to controlling relative entropy and is not accompanied by evidence of resolution.