The binomial-factor conjecture for exponent discrepancies

Define numbers $\omega_m^n$ and $\overline{\omega}_m^n$ by

$$\sum_{d\geqslant0}p_d^nt^d=\prod_{m>0}(1-t^m)^{-\omega_m^n},\qquad \overline{\omega}_m^n=\binom{m+n-3}{n-2},$$

and set $\varepsilon_m^n=\overline{\omega}_m^n-\omega_m^n$.

Exponent-discrepancy factorization conjecture. For every $m\geqslant6$ there exists an irreducible polynomial $r_m(t)\in\mathbb{Q}[t]$ of degree at most $m-6$ such that

$$\varepsilon_m^n=\binom{n}{4}r_m(n)\qquad\text{for all }n\geqslant1.$$

In particular, $\varepsilon_m^n$ has degree at most $m-2$. The claim predicts a substantial degree drop for the error between the correct and predicted exponents, and would imply asymptotic agreement of the corresponding exponent sequences. The source reports computational checks for related claims but gives no resolution of this factorization statement.