The conjecture that the Markoff block action is alternating or symmetric

Let $\mathbb{X}$ be the affine surface defined by

$$x^{2}+y^{2}+z^{2}=xyz.$$

For a prime $p$, let $X^{*}(p)$ be the set of nonzero solutions in $\mathbb{Z}/p\mathbb{Z}$, and partition it into blocks obtained by changing the signs of exactly two coordinates. Let $Y^{*}(p)$ be the set of these blocks, and let $Q_{p}$ be the permutation group induced by the Markoff group $\Gamma$ on $Y^{*}(p)$. The Markoff block-action conjecture. For every $p\geq 5$, the permutation group $Q_{p}$ is the full alternating or symmetric group. This conjecture concerns the structure of the finite permutation groups induced by the Markoff transformations after passing to the sign-change blocks. The source presents it as suggested by simulations and does not report a proof or refutation.