The Strong Four Exponentials Conjecture

Let $\widetilde{\mathcal L}\subset\overline{\mathbf Q}_p$ be the $\overline{\mathbf Q}$-vector space generated by $1$ and the elements of $\log_p(\overline{\mathbf Q}^{\times})$. Let $M$ be a $2\times2$ matrix with entries in $\widetilde{\mathcal L}$, and suppose that its rows and columns are linearly independent over $\overline{\mathbf Q}$. Strong Four Exponentials Conjecture. Then $M$ has rank $2$. This is a $p$-adic transcendence conjecture connected with the expected non-vanishing of $p$-adic regulators; it remains open.