The geodesic growth conjecture for compact Ricci-flat Calabi–Yau manifolds

Let $(X,\tilde{g})$ be a compact, Ricci-flat Calabi–Yau manifold, where a Calabi–Yau manifold means a Kähler manifold $(X,g)$ with vanishing first Chern class $c_1(X)=0$ and trivial first cohomology group $H^1(X;\mathbb{R})=0$. If its real dimension is $n$, let $\mathcal{N}(L)$ denote the relevant counting function for geodesics of length at most $L$. Geodesic growth conjecture. $X$ has stable, closed, non-constant geodesics; in fact, there is a constant $C(X)>0$ such that

$$\mathcal{N}(L)\sim C(X)L^n.$$

The statement is motivated by physics-based arguments of P. Gao and M. Douglas for the Kähler case. Its status is not established in the supplied text.