Sharp vertex connectivity of the Markoff graphs modulo $p$

For a prime $p$ and $k\in\mathbb{F}_p$, the generalized Markoff graph $G_{p,k}$ is an undirected graph whose vertices are the solutions over the finite field $\mathbb{F}_p$ of the normalized Markoff equation \[ x_1^2+x_2^2+x_3^2=x_1x_2x_3+k, \] where two vertices are adjacent if they differ by a Vie