The multipartition equidistribution conjecture

For each positive integer $t$, let $p_t(n)$ be the number of $t$-multipartitions of $n$, defined by

$$\sum_{n=0}^{\infty}p_t(n)q^n=\frac{1}{\prod_{i=1}^{\infty}(1-q^i)^t}.$$

Let $\delta_t$, if it exists, be the density of integers $n$ for which $p_t(n)$ is odd:

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

Multipartition equidistribution conjecture. The density $\delta_t$ exists and equals $1/2$ for every odd positive integer $t$. Equivalently, if $t=2^k t_0$ with $t_0\geq 1$ odd, then $\delta_t$ exists and equals $2^{-k-1}$.

This extends the partition-function conjecture from $t=1$ to multipartition functions and predicts equidistribution modulo $2$ in the odd-$t$ case. The source presents the claim as conjectural and notes that even positivity or existence of these densities is beyond current methods.