Uniqueness conjecture for generalized Markov triples

Let $k\in\mathbb Z_{\geq 0}$. A $k$-generalized Markov number is a number appearing in a $k$-generalized Markov triple. Uniqueness conjecture for generalized Markov triples. For any $k$-generalized Markov number $b$, there is a unique $k$-generalized Markov triple $(a,b,c)$ up to order such that

$$\max\{a,b,c\}=b.$$

This extends the uniqueness question for ordinary Markov numbers to the $k$-generalized setting. The paper proves several special cases, including cases where $b=p$ or $2p$ and certain prime-power cases, while the unrestricted assertion remains open.