Weak $p$-adic Schanuel conjecture

Let $\alpha_1,\ldots,\alpha_n$ be nonzero algebraic numbers. Weak $p$-adic Schanuel conjecture. If $\log_p(\alpha_1),\ldots,\log_p(\alpha_n)$ are linearly independent over $\mathbb{Q}$, then they are algebraically independent over $\mathbb{Q}$. This is a conjectural $p$-adic transcendence statement, stronger than the classical $p$-adic four exponentials conjecture in the context described, and is not established by the results of Brumer, Waldschmidt, and Roy cited in the paper.