Positivity conjecture for the shadow Markoff tree

Let $(A,B,C)$ be an initial triple of solutions associated with

$$(A,B,C)=(1+\alpha\varepsilon,1+\beta\varepsilon,1+\gamma\varepsilon),$$

where $\alpha,\beta,\gamma$ are positive integers and $(\alpha:\beta:\gamma)\in\mathbb{Q}\mathbb{P}^2$. Mutations and permutations act along the Markoff tree, and positivity means that every resulting triple has positive integer entries.

Positivity conjecture. Every triple $(A,B,C)$ of positive integer solutions to the shadow Markoff equation can be obtained from an initial triple with $(\alpha:\beta:\gamma)$ a rational point in the quadrilateral with vertices

$$(0:0:1),\quad (1:0:2),\quad (1:1:1),\quad (0:2:1),$$

via mutations and permutations along the Markoff tree.

This conjecture is intended to give a complete characterization of the positive integer solutions to the shadow Markoff equation, a problem previously formulated for solutions whose ordinary and nilpotent parts are all positive. Its status is unresolved in the supplied source.