Pinned broken-line shelling conjecture for matroid h-vectors

Let $M$ be a matroid. A pinned broken line shelling order is a shelling order obtained from a broken-line shelling whose normal vectors lie in the normal cone of one fixed vertex of the matroid polytope. Let the associated restriction set poset be the poset of restriction sets ordered by inclusion. Pinned broken-line shelling conjecture. There exists a pinned broken line shelling order $<$ such that the restriction set poset coincides with the divisibility poset of a pure multi-complex. This conjecture proposes that the multicomplex realizing the matroid's $h$-vector can arise directly from a shelling order, preserving combinatorial information that generic initial ideals or linear systems of parameters may lose; the source reports computational exploration but no resolution.