The Structural Rank Conjecture for logarithms of algebraic numbers

Let $\mathscr L=\{x\in\mathbf C:e^x\in\overline{\mathbf Q}\}$ be the $\mathbf Q$-vector space of logarithms of algebraic numbers. Let $\mathscr L+\mathbf Q$ be its span with $\mathbf Q$. For a matrix $M$ over a characteristic-zero field, write its entries as $M=\sum_i\ell_iM_i$ in a $\mathbf Q$-basis of the coefficient span, and define its structural rank as the rank over $\mathbf Q(x_1,\ldots,x_r)$ of $M_x=\sum_i x_iM_i$.

Structural Rank Conjecture. For every $M\in M_{m\times n}(\mathscr L+\mathbf Q)$, its rank equals its structural rank.

This conjecture seeks to extend Baker's theorem from linear forms in logarithms to arbitrary matrices of logarithms. Its status is not specified in the source.