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.

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