Schmutz Schaller's systolic disk area conjecture

Let $S_g$ be a closed hyperbolic surface of genus $g$, and let $D_{\operatorname{inj}(S_g)}$ denote an embedded metric disk of radius equal to the injectivity radius of $S_g$. Schmutz Schaller's conjecture. There exists a universal constant $C$ such that

$$\operatorname{Area}(D_{\operatorname{inj}(S_g)})<2\pi\left(3C\pi(g-1)^{2/3}-1\right).$$

This is a reformulation of a conjecture about the systole of a closed hyperbolic surface, relevant here because it would constrain when a disk can be a Cheeger minimizer. Its resolution status is not specified in the source.