Judge–Zanello density-transfer conjecture for multipartition functions

For $t\geq1$, let $p_t(n)$ be the $t$-multipartition function, and let $\delta_t$ denote the density of those $n$ for which $p_t(n)$ is odd, when this limit exists. Density-transfer conjecture. (i) If there is an integer $A\equiv\pm1\pmod6$ with $\delta_A>0$, and $\delta_i$ exists for every $i\leq A$ satisfying $i\equiv\pm1\pmod6$, then $\delta_1>0$. (ii) If there is an integer $A\equiv3\pmod6$ with $\delta_A>0$, and $\delta_i$ exists for every $i\leq A$ satisfying $i\equiv3\pmod6$, then $\delta_3>0$.

This is a corollary of the Judge–Zanello congruence conjecture and asserts that positive odd density transfers from a suitable multipartition function to the partition function or cubic partition function. The paper proves this corollary unconditionally, so the conjecture is solved.