Matsubara-Heo–Telen local-multiplicity conjecture for principal matroid determinants

Let $L\subseteq\mathbb{P}^n$ be a linear subspace of dimension $d$ not contained in any coordinate hyperplane, let $M=\operatorname{M}(L)$, and let $F$ be a flat of $M$. Write $L|_F$ for the restriction associated with $F$, let $(L|_F^{-1})^\vee$ be its projective dual, and let $E_L$ be the principal matroid determinant. Let $L^{-2}$ denote the image of $L^{-1}$ under the coordinate-wise squaring map. Matsubara-Heo–Telen's local-multiplicity conjecture. For each flat $F$ of $M$ such that $(L|_F^{-1})^\vee$ is a hypersurface, the defining polynomial of that hypersurface appears in the factorization of $E_L$ with exponent equal to the local multiplicity of $L^{-2}$ along the stratum indexed by $F$. This conjecture specifies the multiplicities of dual-variety factors in the principal matroid determinant; the source states that the conjectures motivating the paper were left open.