Factorization and dual-degree conjecture for positive powers

Prove that for every integer k\geq 1 and every linear space L\subseteq P^d of dimension d whose matroid M is connected, the generalized matroid determinant E_{L,k}=i^*r^*\operatorname{Chow}(L^k) equals the discriminant \Delta(L^{k+1}), and consequently establish \deg((L^{k+1})^\vee)=(d+1)k^d.

Background

For positive powers, the paper defines E_{L,k}=i^r^\operatorname{Chow}(L^k) and proves a degree formula for E_{L,k}. Computational experiments suggest that, when the associated matroid is connected, E_{L,k} is irreducible and coincides with the defining equation of the dual hypersurface of L^{k+1}.

The paper notes that the zero locus of E_{L,k} contains (L^{k+1})^\vee, but the equality and the resulting dual-degree formula remain conjectural in general. The case k=1 is treated separately and follows from a determinantal expression and matroid irreducibility.