Universal stable 2-systolic inequality in dimension four

Let $X$ be a closed orientable smooth $4$-manifold, and let $\operatorname{stsys}_2(\mathbf{g})$ denote its stable $2$-systole for a Riemannian metric $\mathbf{g}$. Then

Universal stable 2-systolic inequality conjecture. There is a numerical constant $C$, independent of $X$, such that for every metric $\mathbf{g}$ on $X$,

$$\operatorname{stsys}_2(\mathbf{g})^2\leq C\operatorname{vol}_4(\mathbf{g}).$$

The question asks for a universal stable systolic bound across all closed orientable smooth four-manifolds; the source explicitly calls it open.