Quartic Berkovich–Dhar sign-change conjecture

For every integer $n\ge 1$, define $P_n(q)=\prod_{j=1}^{n}(1-q^{3j-2})(1-q^{3j-1})$ and $c_{n,m}=[q^{3m+2}]P_n(q)^4$ for $0\le m\le 4n^2-1$. After all zero coefficients are omitted, the sequence $(c_{n,m})$ in increasing order of $m$ has exactly one sign change, and that change is from positive to negative. The claimed quartic refinement further asserts that, for $n\ge 301$, the transition occurs in an interval of length $3000/n^2$ centered at $\alpha n^2+\beta n+\gamma+\delta/n$, where $\alpha=0.7490800947885107\ldots$ and $\alpha,\beta,\gamma,\delta$ have explicit analytic definitions.