Systole of Riemann surfaces of genus $g$

Question

[Already asked in mathoverflow] In Buser and Sarnak's "On the period matrix of a Riemann surface of large genus", we get $$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log g}\le 2$$ Here $\operatorname{sys}(S)$ is the shortest length of closed noncontractible geodesics in $S$ and $\mathcal{M}_g$ is the moduli space. However, I want to ask if there exists a constant $C>0$ such that $$\liminf_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log g}\ge C\quad ?$$ Any help will be appreciated.