Positive Lyapunov exponent conjecture for projective K3 automorphisms

Let $X$ be a projective $K3$ surface, let $T:X\to X$ be an automorphism with positive topological entropy $h>0$, and fix a Ricci-flat Kähler metric $\omega$. Define the largest Lyapunov exponent of the Lebesgue measure $\mathrm{dVol}$ by

$$\Lambda=\int_X\left(\lim_{N\to+\infty}\frac{h}{2N}\log\|D_xT^N\|_\omega\right)\mathrm{dVol}(x).$$

Positive Lyapunov exponent conjecture. One has $\Lambda>0$.

The exponent is known to be finite and nonnegative. Positivity is identified as a major outstanding problem; the source also notes that a dense $T$-orbit would already be interesting in this setting.