Equality of the Thurston distance and asymmetric Finsler distance for negatively curved metrics

Consider isometry classes of negatively curved metrics with topological entropy equal to $1$. Let $d_T$ be the Thurston distance and let $d_F$ be the distance induced by the asymmetric Finsler norm $\|\cdot\|_T$:

$$d_F(g_1,g_2)=\inf_{\gamma(0)=g_1,\,\gamma(1)=g_2}\int_0^1\|\dot\gamma(t)\|_T\,dt.$$

Distance equality conjecture. The distances $d_T$ and $d_F$ coincide on these isometry classes. The inequality $d_T\leq d_F$ is established, and equality is known on Teichmüller space by Thurston's result, while the asserted equality for negatively curved metrics remains open.