Rank formula via 2-thin, 4-shellable covers in 3D

Show that for every graph G, r_3(G) equals min_{F, C} {|F| + Σ_{X∈C}(3|X| − 6) − Σ_{h∈H(C)}(d_C(h) − 1)}, where the minimum is over all edge subsets F ⊆ E(G) and all 2-thin, 4-shellable vertex covers C of E(G) \ F with each X ∈ C having |X| ≥ 5.

Background

The straightforward 3D generalization of the Lovász–Yemini rank formula fails. Motivated by counterexamples to Dress’s second conjecture, the authors propose restricting to structured covers (2-thin, 4-shellable) that reflect how rigid pieces can overlap.

This conjecture would place r_3 in NP ∩ coNP via such certificates, paralleling results proven for the C_2^1-cofactor matroid.