Strong approximation conjecture for Markoff surfaces

Let $p$ be a prime and let $\mathbf{M}_k$ be the Markoff surface

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

Let $\mathcal{E}_k$ denote the union of the exceptional orbits in $\mathbf{M}_k(\mathbb{F}_p)$, and let $\mathcal{C}_k$ be an orbit. Strong approximation conjecture. For each level $k\ne2$, there is a single orbit $\mathcal{C}_k$ such that

$$\mathbf{M}_k(\mathbb{F}_p)=\mathcal{C}_k\sqcup\mathcal{E}_k.$$

This is the orbit formulation of strong approximation: outside the exceptional orbits, every non-singular level should consist of one large orbit. The source discusses substantial partial results, including a large orbit with a small complement, but does not state that the exact conjecture is resolved.