D--Kakde's Matrix Coefficient Conjecture

Let $\mathscr L=\{x\in\mathbf C:e^x\in\overline{\mathbf Q}\}$ and let $\mathscr L_p$ be the $\mathbf Q$-vector space of $p$-adic logarithms of algebraic numbers. Let $M$ be a square matrix of dimension $n$ with coefficients in either $\mathscr L$ or $\mathscr L_p$.

D--Kakde's Matrix Coefficient Conjecture. If $\det(M)=0$, then there exist nonzero vectors $w,v\in\mathbf Q^n$ such that

$$w^tMv=0.$$

This conjecture asks for a rational vanishing matrix coefficient whenever a square logarithmic matrix is singular. The source states that it is equivalent to the Four Exponentials Conjecture when $n=2$, is implied by the Structural Rank Conjecture, and in its $p$-adic form implies Leopoldt's and Gross--Kuz'min's conjectures.