Bergeron–Ceballos–Küstner's unimodality conjecture for q-Fibonomial coefficients

For $m,n\in\mathbb{N}$, define the $q$-Fibonomial coefficient by

$$\left[\\!\begin{matrix} m+n \\\\ n \end{matrix}\\!\right]_{\mathcal{F}}:=\frac{[F_{m+n}]^!_q}{[F_m]^!_q\\,[F_n]^!_q},$$

where $[F_n]^!_q:=\prod_{k=1}^n[F_k]_q$ and $[n]_q:=1+q+\dots+q^{n-1}$. Bergeron–Ceballos–Küstner's conjecture. The polynomials

$$\left[\\!\begin{matrix}m+n\\\ \end{matrix}\\!\right]_{\mathcal{F}}$$

are unimodal. Bergeron–Ceballos–Küstner introduced these polynomials and conjectured unimodality, while the source proves the conjecture for $n\leq 3$; the general case remains open.