Vanishing criterion for forms with exactly two noncontaining hyperplanes

Prove that for any complexified hyperplane arrangement K, any modular flat F of K, any isomorphism of arrangements ι: H ≅ K/F, and any linearly ordered multiset J of hyperplanes of K, if there exists a proper modular flat F′ of K contained in F such that J contains exactly two hyperplanes not containing F′, then the piecewise algebraic form ω_{K,ι,F,J} = ι^*((\overline{π}_F)_*(ω_{K,J})) on SFM[H] vanishes.

Background

Section 3 defines, for a complexified hyperplane arrangement K in V, a modular flat F, an isomorphism of arrangements ι: H ≅ K/F, and an ordered multiset J of hyperplanes of K, the semi-algebraic form ω{K,ι,F,J} ∈ Ω{PA}(SFM[H], ℝ) via pushforward along the modular projection and pullback along ι.

Lemma \ref{lemmarelform} proves vanishing of ω_{K,ι,F,J} when, for some proper modular flat F′ ⊂ F, the multiset J has fewer than two hyperplanes not containing F′. The authors conjecture the analogous vanishing when J has exactly two such hyperplanes, generalizing a known statement in the braid case.