The conjecture on the asymptotic growth of one-dimensional geodesics

Let $(M,g)$ be a Riemannian or Finsler surface. Let $N_1(T)$ denote the counting function for the one-dimensional objects considered in the preceding discussion, and let $\mathcal F(g)$ be the sum, in $]0,+\infty]$, of the symplectic areas of all equivalence classes of lozenges.

Asymptotic growth conjecture.

$$\lim_{T\to\infty}\frac{N_1(T)}{T^2}=\mathcal F(g).$$

The conjecture proposes an exact quadratic asymptotic governed by the total symplectic area of the equivalence classes. The supplied text does not state whether this formula is known or open.