Zero-Lyapunov-exponent measure conjecture for K3 automorphisms

Let $X$ be a complex K3 surface, let $\Gamma\subseteq \operatorname{Aut}(X)$ be a nonelementary subgroup, and let $\mu$ be a probability measure on a generating set of $\Gamma$. Suppose that $\nu$ is a $\mu$-stationary measure with zero Lyapunov exponents. Zero-Lyapunov-exponent measure conjecture. The support of $\nu$ is not Zariski-dense in $X$. This is known when the group generated by the support of $\mu$ contains parabolic elements; removing that assumption is the stated difficulty. Such a measure is necessarily invariant, and nondense support would force it to be atomic or supported on a finite union of copies of $\mathbb{P}^1(\mathbb{C})$.