The $p$-adic Structural Rank Conjecture

Fix a prime $p$ and let

$$\mathscr L_p=\{\log_p(x):x\in\overline{\mathbf Q}^{\,*}\}\subset\mathbf C_p$$

be the $\mathbf Q$-vector space of $p$-adic logarithms of algebraic numbers. Define structural rank for matrices over a characteristic-zero field by replacing a $\mathbf Q$-basis of the coefficient span with algebraically independent variables, as in the classical case.

$p$-adic Structural Rank Conjecture. For every

$$M\in M_{m\times n}(\mathscr L_p+\mathbf Q)\subset M_{m\times n}(\mathbf C_p),$$

the rank of $M$ equals its structural rank.

This is the natural $p$-adic analogue of the classical Structural Rank Conjecture. The source notes that it has consequences for Leopoldt's and Gross--Kuz'min's conjectures, but gives no resolution.