Gromov's optimal systolic constant conjecture for essential manifolds

Let $M^n$ be an essential closed Riemannian manifold, meaning that $M^n$ is nonsimply connected and there is no continuous map from $M^n$ to a polyhedron of dimension at most $n-1$ inducing an isomorphism on fundamental groups. Write $sys_1(M^n)$ for the length of the shortest non-contractible closed curve and $vol(M^n)$ for its Riemannian volume.

Gromov's systolic conjecture. There exists an absolute constant $C$ such that, for every essential closed Riemannian manifold $M^n$,

$$sys_1(M^n)\leq C\sqrt{n}\,vol^{\frac{1}{n}}(M^n).$$

The conjecture asks whether the optimal dimension-dependent constant in Gromov's systolic inequality has order at most $\sqrt n$, as suggested by round real projective spaces and certain flat tori. The exact assertion that round projective spaces attain the optimum is stronger; the weaker constant-factor formulation remains the stated conjecture here.