The Matrix Coefficient Conjecture for singular matrices of logarithms

Let $\mathscr{L}$ be the $\mathbf{Q}$-vector space of logarithms of algebraic numbers, and for a prime $p$ let $\mathscr{L}_p$ be the corresponding space of $p$-adic logarithms. Let $M$ be a square matrix over one of these spaces. Matrix Coefficient Conjecture. If

$$M \in M_{n \times n}(\mathscr{L}) \text{ or } M_{n \times n}(\mathscr{L}_p), \qquad \det(M)=0,$$

then there exist nonzero vectors $w,v\in\mathbf{Q}^n$ such that

$$\langle w,Mv\rangle=0.$$

The conjecture asserts that a singular square matrix of logarithms has a vanishing coefficient after rational changes of basis on the left and right. The paper presents it as weaker than the Structural Rank Conjecture, while explaining that it would imply major open problems including Four Exponentials, Leopoldt, and Gross–Kuz'min; its resolution status is not specified in the supplied text.