Homology preservation in Λ(Δ), the induced subcomplex on all vertices with nontrivial link homology

Prove that for every simplicial complex Δ and every integer j ≥ −1, the reduced homology groups H_j(Λ(Δ)) and H_j(Δ) are isomorphic, where Λ(Δ) is the induced subcomplex of the barycentric subdivision B(Δ) on vertices u_σ with h(Δ,σ) ≠ ∅ (i.e., σ has a link in Δ with nontrivial homology).

Background

As a broader version of Conjecture 7.9, Section 7.3 introduces Λ(Δ), formed by collecting all barycentric vertices whose links in Δ have some nontrivial homology, and conjectures that this subcomplex preserves the full homology of Δ.

Since Λ(Δ) coincides with Λ_i(Δ) for sufficiently large i, proving Conjecture 7.10 would establish that one canonical subcomplex of B(Δ) captures all homological information of Δ.