The adelic Tau conjecture

For $f\in\mathbb{Z}[x_1]$, let $\tau(f)$ denote its number of nonzero coefficients, and let the roots of $f$ be counted in the indicated fields. Adelic Tau conjecture. There is an absolute constant $c$ such that, for any $f\in\mathbb{Z}[x_1]$, there is a field

$$L\in\{\mathbb{R},\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$$

such that $f$ has no more than $(\tau(f)+1)^c$ roots in $L$. This generalizes the Tau conjecture by asserting that one of the real or $p$-adic fields must provide a polynomial bound on the number of roots in terms of the sparsity; the examples immediately preceding it motivate the simultaneous adelic formulation, which remains open.