Markov polynomial weak log-concavity conjecture

Let $M_\rho$ be a Markov polynomial, and let its coefficients be regarded as weights on the corresponding Newton polygon. A sequence is log-concave when $x_k^2\geq x_{k-1}x_{k+1}$ for every interior index $k$. The weights satisfy weak log-concavity when their sequences in the horizontal, vertical, and diagonal directions with $i+j$ constant are all log-concave. Log-concavity conjecture. The coefficients of Markov polynomials are weakly log-concave on the corresponding Newton polygon. This extends the observed coefficient-positivity phenomenon to a directional log-concavity property; the source reports experimental evidence but no proof or resolution.