The nonexistence of simple parity congruences for even-regular partitions when 3 divides the odd part

Let $m_0$ be the odd part of an even integer $m$, and let $b_m(n)$ denote the coefficients of the corresponding $m$-regular partition generating function. Theorem gives congruences between subprogressions of $b_m$ and multipartition functions $p_t$. Nonexistence conjecture. No congruence of the simple form given in Theorem holds when $m_0\equiv 0\pmod{3}$. This predicts that the displayed family does not extend to the case where the odd part of the regularity parameter is divisible by $3$; the source gives no resolution of this conjecture.