Cantat–McMullen Kummer rigidity conjecture for K3 automorphisms

Let $X$ be a $K3$ surface, let $T:X\to X$ be an automorphism with positive topological entropy, let $\mathrm{dVol}=\Omega\wedge\overline{\Omega}$ be the Lebesgue measure, and let $\mu$ be the measure of maximal entropy. Cantat–McMullen's Kummer rigidity conjecture.

$$\mu\ll \mathrm{dVol}\quad\Longleftrightarrow\quad (X,T)\text{ is a Kummer example}.$$

The conjecture asserts that absolute continuity of the maximal-entropy measure characterizes Kummer examples; in that case one also has $\mu=\mathrm{dVol}$. It is presented as an analogue of rigidity results for Lattès maps.