Divisibility conjecture for generic Markoff-surface orbits

Let $p$ be a prime, and let

$$\mathbf{M}_k:x^2+y^2+z^2-xyz-2=k$$

be a Markoff surface over $\mathbb{F}_p$. Call a level $k$ generic when it contains no orbits associated with $A_4$, $S_4$, or $A_5$. Let $\mathcal{O}$ be an orbit at a generic level $k$, and suppose that $\mathcal{O}$ is not exceptional, meaning that it contains a triple with at least two nonzero coordinates. Divisibility conjecture. The orbit size satisfies

$$|\mathcal{O}|\text{ is divisible by }p-\left(\frac{k^2-4}{p}\right).$$

Here $(\frac{\cdot}{p})$ is the Legendre symbol. The conjecture is motivated by the expected sizes of the large, or cage, orbits after exceptional orbits are removed; the source gives no resolution status.