Loewner's bound for aspherical surfaces

Let $\Sigma$ be a closed aspherical surface, equivalently a closed surface with infinite fundamental group and non-positive Euler characteristic, excluding $S^2$ and $\mathbb{R}\mathbb{P}^2$. Write $\operatorname{sys}(\Sigma)$ for its systole and $\operatorname{area}(\Sigma)$ for its area.

Loewner's bound. Every aspherical surface satisfies

$$\frac{\operatorname{sys}(\Sigma)^2}{\operatorname{area}(\Sigma)}\leq\frac{2}{\sqrt{3}}.$$

This is a systolic-geometric extension of Loewner's torus inequality to all closed aspherical surfaces. The source says that the conjecture has been discussed in the systolic literature, but supplies no resolution here.