Neguț's positivity conjecture for generalized $q,t$-Catalan polynomials

Let $a_1\geqslant a_2\geqslant \cdots\geqslant a_n\geqslant 0$, and let $f(a_1,\ldots,a_n)$ be the polynomial in $q$ and $t$ defined in the paper.

Neguț's conjecture. The polynomial $f(a_1,\ldots,a_n)$ has nonnegative coefficients.

This conjecture generalizes the coefficient positivity of rational and ordinary $q,t$-Catalan numbers to arbitrary weakly decreasing sequences. It is proved in the paper for the cases covered by its results, but remains open in general.