Non-vanishing conjecture for principal specializations of monomial symmetric polynomials

Let $n$ be a prime power and let $k$ be a positive integer. Let $\Lambda_n^k$ be the indexing set of partitions used in the paper, let $|\lambda|$ denote the size of $\lambda$, let $m_\lambda$ be the monomial symmetric polynomial indexed by $\lambda$, and let $\zeta_{(n,k)}$ be the corresponding specialization. Also let $\Theta(\mathbb{Z}/n\mathbb{Z})$ and $\tilde{\Lambda}_n^k$ be the group-determinant and indexing-set objects defined in the paper. Non-vanishing conjecture. For any $\lambda\in\Lambda_n^k$, where $n$ is a prime power and $k$ is a positive integer,

$$|\lambda|\equiv 0\pmod n$$

if and only if

$$m_\lambda(\zeta_{(n,k)})\neq 0.$$

Equivalently, for any positive integer $k$, $n$ is a prime power if and only if

$$\operatorname{N}\left(\Theta(\mathbb{Z}/n\mathbb{Z})^k\right)=|\tilde{\Lambda}_n^k|.$$

The conjecture generalizes the paper's established non-vanishing result for $m_\lambda(\zeta_{(p,1)})$ when $p$ is prime. Necessary and sufficient conditions for non-vanishing are otherwise not known, including cases related to Theorems 1.1(2) and 1.2(1)--(4).