Weak $p$-adic four exponentials conjecture

Let $\mathbb{L}$ be the $\overline{\mathbb{Q}}$-linear subspace of $\overline{\mathbb{Q}}_p$ generated by $1$ and the $p$-adic logarithms of nonzero algebraic numbers. Let $N$ be a $2\times 2$ matrix with entries in $\mathbb{L}$, and suppose that all four entries are $\overline{\mathbb{Q}}$-linearly independent. Weak $p$-adic four exponentials conjecture. Then $N$ has rank $2$. The conjecture is presented as a variant of the $p$-adic four exponentials conjecture; it does not follow from the stated rank bound of Roy and Waldschmidt, but is much weaker than the weak $p$-adic Schanuel conjecture.