Local unmarked length spectrum rigidity for hyperbolic surfaces

Let $(M,g)$ be a closed negatively curved surface. If $g$ is strictly $\frac{1}{9}$-pinched and has the same unmarked length spectrum as a hyperbolic metric, we show that $g$ is hyperbolic. As a consequence, we show that any hyperbolic metric on a surface admits a $C^2$-neighborhood in the space of metrics in which it is characterized by its unmarked length spectrum, up to isometry.