The systolic-volume bound for KVol on translation surfaces

Let $X$ be a translation surface. Write $\mathop{\mathrm{KVol}}(X)$ for its KVol and $\mathop{\mathrm{SysVol}}(X)$ for its systolic volume, defined as the volume divided by the square of the homological systolic length. The systolic-volume conjecture.

$$\mathrm{KVol}(X) \leq \frac{2}{\sqrt{3}}\mathrm{SysVol}(X).$$

This would sharpen the general Riemannian-surface bound $\mathrm{KVol}(X)\leq 9\mathrm{SysVol}(X)$ and provide a uniform comparison for translation surfaces, although the preceding inequality with constant $1$ is known to fail in general.