Unimodality conjecture for the $q$-factorial quotient

For $n\geqslant 2$, define

$$P_n(q)=\frac{[6n]![n]!}{[3n]![2n]!^2},$$

where $[m]!=(1-q)(1-q^2)\cdots(1-q^m)$. Unimodality conjecture. The polynomial $P_n(q)$ is unimodal. The polynomial is known to be reciprocal, and the conjectured unimodality would imply non-negative coefficients for two polynomials in Theorem 2; the supplied text gives no proof or disproof of this conjecture.