Chu–Herzog–Lu conjecture on socle ideals of polymatroidal ideals

Let $K$ be a field, $S=K[x_1,\dots,x_n]$, and let $I\subset S$ be a polymatroidal ideal. The socle ideal $\operatorname{soc}(I)$ is the unique monomial ideal satisfying $I:(x_1,\dots,x_n)=\operatorname{soc}(I)+I$. Chu–Herzog–Lu's conjecture. The socle ideal $\operatorname{soc}(I)$ is polymatroidal. A positive answer to the Bandari–Bayati–Herzog conjecture would imply this conjecture through the highest homological shift ideal. It is known for several classes of polymatroidal ideals, including ideals generated in degree two, ideals in at most three variables, matroidal ideals, principal Borel ideals, PLP-polymatroidal ideals, and LP-polymatroidal ideals; the general case remains open.