The Markoff graph connectivity conjecture modulo primes

Let $\mathcal M$ be the set of positive integer solutions to

$$x^2+y^2+z^2=3xyz,$$

and let $\mathcal M_p$ be the set of non-zero solutions modulo a prime $p$. The functional graph is generated by the involutions

$$\mathcal R_1(x,y,z)=(3yz-x,y,z),$$

and the analogous maps $\mathcal R_2,\mathcal R_3$, together with permutations of the coordinates. Let $\mathcal C_p\subseteq\mathcal M_p$ denote the largest connected component of the resulting graph. Baragar–Bourgain–Gamburd–Sarnak conjecture. For every prime $p$, one has

$$\mathcal C_p=\mathcal M_p.$$

This asserts that the Markoff triples generate all non-zero solutions modulo $p$ and that the associated functional graph is connected; the supplied text gives no evidence resolving the conjecture.