The bounded-exception conjecture for Markoff surface orbits

Let $a,k\in\mathbb Z$, and let

$$\mathcal M_{a,k}: x^2+y^2+z^2=axyz+k$$

be the Markoff surface over $\mathbb Z$, with $\mathcal G_{\mathcal M}$ the group generated by the three involutions, double sign changes, and coordinate permutations. Bounded-exception conjecture. (a) There is a constant $M_1(a,k)$ such that for every prime $p\nmid a$,

$$\#\mathcal M_{a,k}(\mathbb F_p)\leq \#\bigl(\text{largest $\mathcal G_{\mathcal M}$-orbit in $\mathcal M_{a,k}(\mathbb F_p)$}\bigr)+M_1(a,k).$$

(b) If $\#\mathcal M_{a,k}(\mathbb Z)=\infty$, then there is a constant $M_2(a,k)$ such that for every prime $p\nmid a$,

$$\#\mathcal M_{a,k}(\mathbb F_p)\leq \#\bigl(\mathcal M_{a,k}(\mathbb Z)\bmod p\bigr)+M_2(a,k).$$

One may further ask whether $M_1(a,k)$ and $M_2(a,k)$ can be chosen independently of $a$ and $k$. The source presents these assertions as conjectures motivated by finite orbits and lifting questions; no resolution is given.