$\ell$-adic Schanuel conjecture

Let $\bar{\mathbb Q}_\ell$ be an algebraic closure of $\mathbb Q_\ell$, and let $\log$ denote the Iwasawa $\ell$-adic logarithm. Let $x_1,\ldots,x_n\in\bar{\mathbb Q}_\ell^\times$ be algebraic over $\mathbb Q$. Assume that $\ell,x_1,\ldots,x_n$ are multiplicatively independent, meaning that

$$\ell^c\prod_{i=1}^n x_i^{c_i}\neq 1$$

for every nonzero $(c,c_1,\ldots,c_n)\in\mathbb Z^{n+1}$.

$\ell$-adic Schanuel conjecture. The numbers $\log x_1,\ldots,\log x_n$ are algebraically independent over $\mathbb Q$.

This is the particular case of the $\ell$-adic Schanuel conjecture used in the paper. It implies the main regulator nonvanishing conjecture, but is itself unproved.