Judge–Keith–Zanello multipartition-density conjecture

Let $p_t(n)$ be the $t$-multipartition function, and, when the limit exists, define the density of its odd values by

$$\delta_t=\lim_{x\to\infty}\frac{\#\{n\leq x:p_t(n)\text{ is odd}\}}{x}.$$

Judge–Keith–Zanello conjecture. The density $\delta_t$ exists and equals $1/2$ for every positive odd integer $t$. Equivalently, if $t=2^k t_0$ with $t_0\geq1$ odd, then $\delta_t$ exists and equals $2^{-k-1}$.

It generalizes the Parkin–Shanks conjecture from the partition function to all multipartition functions. At present, existence of any of these densities is not known in general, so the conjecture remains open.