The adelic root bound hypothesis for sums of products of sparse polynomials

For $k,m,t\in\mathbb{N}$, let $\mathrm{SPS}(k,m,t)$ denote the class of polynomials represented as sums of $k$ products of $m$ integer polynomials, each having at most $t$ nonzero terms. A distinct root is a root counted without multiplicity. Adelic root bound hypothesis. For any $k,m,t\in\mathbb{N}$ and $f\in\mathrm{SPS}(k,m,t)$, 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 $(kmt)^{O(1)}$ distinct roots in $L$. This is an open problem related to $p$-adic and real-analytic approaches to bounding integer roots. A stronger assertion, that every integer polynomial has only $\tau(f)^{O(1)}$ roots in $L$, is known to be false.