Prime cyclotomic factor conjecture for unimodal generating functions

Let $\Phi^{\operatorname{uni}}$ denote the monoid of cyclotomic generating functions that are unimodal, and let $\Phi_p(q)$ denote the $p$th cyclotomic polynomial for a prime $p$. Prime cyclotomic factor conjecture. Every nontrivial element $f\in\Phi^{\operatorname{uni}}$ has a cyclotomic factor of the form $\Phi_p(q)$ for some prime $p$.

This is proposed as an analogue of the corresponding factorization property for positive cyclotomic generating functions, motivated by the apparent smoothing effect of multiplication by $\Phi_p(q)$. It had been checked computationally through degree $50$ at the time of writing, but no general proof is given.