Nontriviality of the Nori fundamental group of fake projective planes in positive characteristic

Let $X$ be a fake projective plane over an algebraically closed field of characteristic $p>0$, meaning a smooth projective surface with the same Betti numbers as the projective plane and satisfying $q=0$, $p_g=0$, and $K_X^2=9$. Let $\pi_1^{Nori}(X)$ denote its Nori fundamental group scheme. Nontriviality conjecture. One should have

$$\pi_1^{Nori}(X)\neq 1.$$

In characteristic zero, the corresponding nontriviality follows from the preceding argument, whereas the positive-characteristic assertion is presented as a conjectural extension and is not resolved here.