Two-sided majorization conjecture for cyclotomic generating functions

Let

$$f(q)=\prod_{k=1}^m \frac{[a_k]_q}{[b_k]_q}\in\Phi^{+},$$

where $a_1\leq\cdots\leq a_m$ and $b_1\leq\cdots\leq b_m$. Two-sided majorization conjecture. For every $\ell$, one has

$$\sum_{k=1}^{\ell}a_k\geq\sum_{k=1}^{\ell}b_k \qquad\text{and}\qquad \sum_{k=\ell}^{m}a_k\geq\sum_{k=\ell}^{m}b_k.$$

Equivalently, the multiset of numerator parameters weakly majorizes the multiset of denominator parameters from both sides. The conjecture strengthens the partial inequalities established earlier in the paper and has implications for the structure of positive cyclotomic generating functions.