The saturation conjecture for Markov polynomials

Let $\rho=\frac{a}{b}\frac{}{}$ be a rational number in $[0,1]$, and write the numerator of its Markov polynomial as

$$P_{\rho}(u,v,w)=\sum A_{ij}u^iv^jw^{a+b-1-i-j}.$$

Its Newton polygon is

$$\Delta_{\rho}=\operatorname{Conv}\lbrace(i,j):A_{ij}\neq 0\rbrace.$$

Saturation conjecture. The terms that appear in the numerator of a Markov polynomial $M_{\rho}$ are precisely those corresponding to the integer lattice points in the Newton polygon $\Delta_{\rho}$; equivalently, every point of $\Delta_{\rho}\cap\mathbb{Z}^2$ corresponds to a monomial with nonzero coefficient in $P_{\rho}$.

This conjecture asserts that the support of each Markov polynomial saturates its Newton polygon at all integer lattice points. The surrounding text attributes the Newton-polygon description and the conjecture to earlier work, but no resolution status is supplied here.