Factor 4 conjecture for Markov-polynomial coefficients

Let $\rho=a/b$ be rational, and let the critical triangle be the set of lattice points satisfying

$$i<a,\qquad j<b,\qquad \frac{i}{a}+\frac{j}{b}>1.$$

Consider the coefficients of the Markov polynomial associated with these lattice points. Factor 4 conjecture. Every coefficient of a Markov polynomial corresponding to a lattice point strictly within the critical triangle is divisible by $4$. The conjecture is based on experimental data about coefficients in the critical-triangle region; the source gives no resolution.