The strong approximation conjecture for the Markoff graph

Let $\mathcal{M}$ be the set of Markoff triples over $\mathbb{Z}$, let $\mathcal{M}_p$ be the set of their reductions modulo a prime $p$, and let $\mathcal{G}^\times_p$ be the Markoff graph modulo $p$. The canonical projection identifies triples congruent modulo $p$. Strong approximation conjecture. For every prime $p$, the Markoff graph $\mathcal{G}^\times_p$ is connected and is equivalent to $\mathcal{G}^\times$ with triples congruent modulo $p$ identified; equivalently, the canonical projection $\mathcal{M}\to\mathcal{M}_p$ is onto. This conjecture asserts that every solution to the Markoff equation modulo a prime lifts to a Markoff triple over $\mathbb{Z}$. Strong approximation is known for all but finitely many primes, but remains open in general; it is a first step toward the expander-family conjecture for these graphs.