Strong Approximation Conjecture for the Markoff surface

Let $\mathbb X$ be the affine surface

$$\mathbb X: x_1^2+x_2^2+x_3^2-3x_1x_2x_3=0.$$

Let $\Gamma$ be the group of affine integral morphisms of $\mathbb A^3$ generated by coordinate permutations and the Vieta involutions. For a prime $p$, let $\mathbb X(\mathbb Z/p\mathbb Z)$ be the set of solutions modulo $p$, and define

$$X^*(\mathbb Z/p\mathbb Z)=\mathbb X(\mathbb Z/p\mathbb Z)\setminus\{(0,0,0)\}.$$

Strong Approximation Conjecture. For any prime $p$, $\mathbb X(\mathbb Z/p\mathbb Z)$ consists of exactly two $\Gamma$-orbits, namely

$$\{(0,0,0)\}\quad\text{and}\quad X^*(\mathbb Z/p\mathbb Z).$$

This asserts that the nonzero solutions to the Markoff equation form a single orbit under the natural Vieta and permutation symmetries, a strong approximation property for the Markoff surface. The supplied text does not state whether the conjecture is proved or remains open.