Markov polynomial Newton-polygon saturation conjecture

Let $\rho=a/b$ be a rational number, and let $\Delta_{a/b}$ be the Newton polygon of the numerator of the Markov polynomial $M_\rho(x,y,z)$, consisting of the integer points $i,j\geq0$ satisfying

$$\frac{i}{a}+\frac{j}{b}\geq1,\qquad i+j\leq a+b-1.$$

Saturation conjecture. The terms appearing in the numerator of $M_\rho$ are precisely those corresponding to the integer lattice points in $\Delta_\rho$. The conjecture is an equivalent explicit formulation of the assertion that all coefficients associated with lattice points of the Newton polygon are positive; the source gives no resolution.