Markov polynomial classification conjecture for Laurent-polynomial solutions

Let $X,Y,Z\in\mathbb Z[x^{\pm1},y^{\pm1},z^{\pm1}]$ be integer Laurent polynomials satisfying equation

. Markov polynomials $M_\rho(x,y,z)$ are the Laurent polynomials associated with rationals $\rho$ through the Frobenius parametrization, and triples of Markov polynomials are the corresponding Markov-polynomial triples. **Markov polynomial classification conjecture.** Up to changing the signs of any two of $X,Y,Z$, every such solution is a triple of Markov polynomials. This would classify all integer Laurent-polynomial solutions of

in terms of the Markov-polynomial construction; the source gives no resolution.