Eventual unimodality of the coefficients of the q-Catalan numbers

Let $C_n(q)=\sum_{k=0}^{n(n-1)}m_n(k)q^k$ denote the $q$-Catalan numbers and let $m_n(k)$ be their coefficients. Eventual unimodality conjecture. The sequence

$$\{m_n(1),\ldots,m_n(n(n-1)-1)\}$$

is unimodal when $n$ is sufficiently large. The limiting normal distribution suggests that the coefficients become almost unimodal after omitting the first and last terms; the claim is supported by computation showing that $C_n(q)$ is unimodal for $n\geq 16$, but remains unproved.