Strong Approximation for the Markoff surface modulo primes

Let $\mathcal{X}$ be the affine surface in $\mathbb{A}^3$ defined by

$$X_1^2+X_2^2+X_3^2=3X_1X_2X_3,$$

and let $\Gamma$ be the group of affine integral morphisms generated by coordinate permutations and the Vieta involutions, including

$$R_1(x_1,x_2,x_3)=(3x_2x_3-x_1,x_2,x_3),$$

with $R_2$ and $R_3$ defined similarly. Strong Approximation. For any prime $p$,

$$\Gamma\cdot(1,1,1)=\mathcal{X}(\mathbb{Z}/p\mathbb{Z})-\{(0,0,0)\}.$$

This asserts that every nonzero solution of the Markoff equation modulo a prime lies in the orbit of $(1,1,1)$. Bourgain, Gamburd, and Sarnak proved the assertion for primes $p$ such that $p^2-1$ does not have too many divisors; the general statement remains open.