McCullough–Wanderley Q-classification conjecture for essential Markoff triples

Let $q$ be a prime power and let an essential triple $(x,y,z)\in\mathbb{F}_q^3$ be the image under $(A,B)\mapsto(\operatorname{tr}A,\operatorname{tr}B,\operatorname{tr}(AB))$ of a generating pair $A,B\in\operatorname{SL}_2(\mathbb{F}_q)$. Markoff equivalence is generated by Vieta involutions and coordinate permutations, and a Markoff class is an equivalence class for this relation. $Q$-classification conjecture. The Markoff class of an essential triple $(x,y,z)\in\mathbb{F}_q^3$ is uniquely determined by $\kappa\coloneqq x^2+y^2+z^2-xyz$. This is the Markoff-triple formulation of the classification problem; the supplied source gives no resolution.