Rigidity conjecture for Lyapunov-exponent maximizers on the two-torus

Let $f$ be a $C^2$ volume-preserving Anosov diffeomorphism of $\mathbb{T}^2$, and let $\Lambda:\operatorname{Diff}^2_m(\mathbb{T}^2)\to\mathbb{R}$ be the functional defined in the source. Rigidity conjecture. If $f$ is a local maximum for $\Lambda$, then $f$ is $C^1$ conjugate to a linear toral automorphism. This is a rigidity statement for local maximizers of a Lyapunov-exponent functional; the supplied text gives no evidence that it has been proved or disproved.