Gatzouras–Peres conjecture on measures of maximal dimension

Let $T:M\rightarrow M$ be an expanding map, and let $K\subseteq M$ be a compact invariant set satisfying the specification property. A $T$-invariant measure has the same Hausdorff dimension as $K$ when its Hausdorff dimension equals that of $K$.

Gatzouras–Peres conjecture. There exists a unique ergodic $T$-invariant measure with the same Hausdorff dimension as $K$. Moreover, this measure is mixing for $T$ and is possibly measurably isomorphic to a Bernoulli shift.

The conjecture concerns the existence and statistical properties of a distinguished measure realizing the Hausdorff dimension of an invariant set in an expanding dynamical system. The supplied text gives no resolution status beyond attributing the conjecture to D. Gatzouras and Y. Peres.