NP-hardness conjecture for p-adic feasibility of sparse polynomials

For a prime $p$, let ${\mathcal F}^*_{n,n+1}$ denote the class of $(n+1)$-nomials in $n$ variables, and let ${\text{{\tt FEAS}}}_{\mathbb{Q}_p}$ be the feasibility problem for rational roots over the $p$-adic field $\mathbb{Q}_p$. The p-adic NP-hardness conjecture. For any fixed prime $p$,

$${\text{{\tt FEAS}}}_{\mathbb{Q}_p}\!\left(\bigcup_{n\in\mathbb{N}} {\mathcal F}^*_{n,n+1}\right)$$

is ${\mathbf{NP}}$-hard. The analogous real feasibility problem is already known to be ${\mathbf{NP}}$-hard for broader families with more than $n+1$ monomials, and the conjecture predicts the same phenomenon for multivariate $(n+1)$-nomials over $\mathbb{Q}_p$.