The entropy threshold conjecture for cat-map quantum limits

Let $A\in SL(2,\mathbb{Z})$ be a hyperbolic cat map, let $\mu$ be a quantum limit for $A$, let $\operatorname{ord}_N(A)$ be the order of $A$ modulo $N$, and let $h_{\mathrm{max}}$ denote the maximal entropy. Suppose that

$$\liminf_{N\to\infty}\frac{\operatorname{ord}_N(A)}{\log N}\geq C.$$

Entropy threshold conjecture. The entropy of $\mu$ satisfies

$$h(\mu)\geq h_{\mathrm{max}}-\frac{1}{C}.$$

In particular, if

$$\liminf_{N\to\infty}\frac{\operatorname{ord}_N(A)}{\log N}=\infty,$$

then $\mu$ is Lebesgue measure.

This conjecture predicts the transition from the logarithmic-scale non-QUE examples for cat maps to QUE just beyond that scale. It would improve the known QUE results at polynomial propagator-order scales and quantify how short periods constrain entropy.