Baragar's conjecture on connectivity of Markoff graphs

Let $p$ be a prime, and let $\mathcal{G}_p$ be the Markoff graph over $\mathbb{F}_p$: its vertices are the nonzero triples in $\mathbb{F}_p^3$ satisfying

$$x_1^2+x_2^2+x_3^2=3x_1x_2x_3,$$

and two vertices are joined when one is obtained from the other by a Vieta involution. Baragar's conjecture. The graph $\mathcal{G}_p$ is connected for any prime $p$. This is equivalent to the assertion that every solution modulo $p$ of the Markoff equation lifts to an integral Markoff triple. Chen's theorem, together with work of Bourgain, Gamburd, and Sarnak, proves the conjecture for all but finitely many primes; the statement for every prime is the conjectural form.