Markov Sail Duality conjecture for Markov-polynomial coefficients

Let $\mathcal M(C)$ be the coefficient of the monomial represented by an interior integer point $C$ of the critical triangle of a Markov sail. Along an unbroken line segment between vertices $C_i$ and $C_{i+1}$, let $d(C_iC_{i+1})$ be the common difference of the $\mathcal M$-values of the integer points on that segment. For the sail vertices $A_i$ and $B_i$, the conjectured relations are

$$d(B_iB_{i+1})=-\mathcal M(A_{i+1}),$$ $$d(A_{i+1}A_{i+2})=-\mathcal M(B_{i+1}).$$

Markov Sail Duality conjecture. The coefficients of Markov polynomials satisfy these two Markov-sail duality identities. This would determine almost all interior sail coefficients from an initial value together with the integer lengths of unbroken lines; the source gives no resolution.