The homotopical 2-systole conjecture

Let $(M,g)$ be a closed Riemannian manifold with scalar curvature $R(g)\geq 2$, and suppose that its universal covering is homotopic to the $2$-sphere. Define the homotopical $2$-systole by

$$\operatorname{sys}_2(M,g)=\inf\left\{\operatorname{area}(\mathbf S^2,i^*g)\mid i:\mathbf S^2\to M\text{ is smooth and }[i]\text{ is homotopically nontrivial}\right\}.$$

Homotopical $2$-systole conjecture. Under these hypotheses,

$$\operatorname{sys}_2(M,g)\leq 4\pi.$$

The conjecture is motivated by the sharp estimate proved in the paper for dimensions $2\leq n\leq 5$ under a nonzero-degree domination hypothesis whose target has universal-cover homology vanishing in degrees at least three. The supplied text does not state whether the conjecture has been resolved in general.