Systolic lower-bound conjecture for Hofer–Zehnder capacity

Let $M^n$ be an essential closed Riemannian manifold of dimension $n$. Let $\langle \pi_1(M)\rangle'$ denote the set of non-trivial conjugacy classes of the fundamental group, and let $DM$ be the unit disc bundle of $M$. The quantity $C_{HZ}(DM,\langle \pi_1(M)\rangle')$ is the Hofer–Zehnder capacity sensitive to these non-trivial classes.

Systolic capacity conjecture. There is a constant $D_n$ depending only on $n$ such that

$$(\operatorname{Vol} M)^{\frac 1n} \leqslant D_n \cdot C_{HZ}(DM,\langle \pi_1(M)\rangle').$$

The conjecture is motivated by the relation between Hofer–Zehnder capacities and systolic constants, together with Gromov's systolic inequality. Its resolution is not specified in the supplied material.