Logarithmic relative-systole alternative for quasi-optimal pseudomanifolds

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing $n\,\mathbf a$, and let $\varphi_n:\pi_1(P_n)\to\pi_1(X)$ be the induced homomorphism. Write $\operatorname{sys}_{\varphi_n}(P_n)$ for the relative systole and $\sigma_{\varphi_n}(P_n)$ for the corresponding relative systolic volume. Logarithmic relative-systole alternative. One should have either

$$\operatorname{sys}_{\varphi_n}(P_n)\gtrsim\log n$$

or

$$\sigma_{\varphi_n}(P_n)\lesssim\frac{n}{\log^m n}.$$

This alternative is proposed after deriving an entropy estimate conditional on the local volume-growth conjecture. It is a heuristic direction in the paper, and no resolution is supplied.