Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy classes of closed curves in $S$. The marked length spectrum function is $l_{g_i}: \mathcal{C} → \mathbb{R}^{+}$ which assigns to the class $[\gamma]$ the length $l_{g_i}(\gamma)$ of the closed geodesic in $[\gamma]$.

My question is that given the curvature condition $K_{g_1}$ $<$ $K_{g_2}\leq -1$, can one conclude that $$l_{g1}([\gamma])\leq l_{g_2}([\gamma]),$$ for all $[\gamma]\in\mathcal {C}$? i.e. the length of a closed geodesic in $(S,g_2)$ is longer then the length of the corresponding closed geodesic in $(S,g_1)$.

**p.s. the converse is not true. Thanks to the answer from @ Igor Rivin and @Anton Petrunin.**

Thanks for the help.