q-Markov function conjecture for mirror Markov-tree branches

Fix a Markov number $\ell\geq 5$. Let $m_\ell(q)$ be a mirror deformation of $\ell$, and let $(p_n)_{n\geq 0}$ be the sequence of polynomials on a mirror Markov-tree branch fixing $m_\ell$, with $p_0=m_\ell$. Define

$$M=m_\ell(q)m_\ell(q^{-1}),\qquad P_n=p_n(q)p_n(q^{-1}).$$

For the branch $(m_\ell,p_{n-1},p_n)$, let $\varepsilon=1$ for a red branch and $\varepsilon=-1$ for a blue branch, and set $d=\deg(p_{n+1})+\varepsilon$. q-Markov function conjecture. There exists a possibly different $q$-deformation $a_\ell(q)\in\mathbb{Z}_{\geq 0}[q]$ of $\ell$ such that

$$\mathcal{M}_q^{\ell}(m_\ell,p_{n-1},p_n)=\frac{q^{d}M+q^{d+\varepsilon}P_n+q^{d-\varepsilon}P_{n-1}}{a_\ell p_np_{n-1}}$$

is constantly equal to $[3]_q$. This is presented as an open question, with only computational evidence and no stated resolution.