Eventual log-concavity 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. Log-concavity conjecture. There exists an integer $t$ such that, when $n$ is sufficiently large, the sequence

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

is log-concave, namely

$$\left(m_n(k)\right)^2\geq m_n(k+1)m_n(k-1)$$

for $t+1\leq k\leq n(n-2)-t-1$. Moreover, the minimum value of $t$ seems to be $75$. This is presented as a stronger conjecture suggested by numerical evidence for $n>70$; it remains open.