The algebraic Matrix Coefficient Conjecture for matrices of logarithms

Let $\mathscr{L}$ be the $\mathbf{Q}$-vector space of logarithms of algebraic numbers, and let $\mathscr{L}_p$ be its $p$-adic analogue. Consider an $n\times n$ matrix whose entries lie in the $\overline{\mathbf{Q}}$-vector space spanned by one of these spaces. Algebraic Matrix Coefficient Conjecture. If

$$\det(M)=0,$$

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

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

This is the $\overline{\mathbf{Q}}$-version of the Matrix Coefficient Conjecture. The source notes that a suitably strong Waldschmidt–Masser result would imply it and poses as an open problem whether the rational version, or a suitable strengthening remaining over $\mathbf{Q}$, implies this algebraic version.