Direct-sum decomposition formula for S^2_{0,0}(X;α) in terms of cable Khovanov homology

Prove the direct-sum formula asserting that, for a 2-handlebody X obtained by attaching 2-handles along a framed oriented m-component link K ⊂ S^3 with diagram D having no 0-framed unknot components, and for any α ∈ Z^m, the Khovanov skein lasagna module S^2_{0,0}(X;α) is isomorphic to the direct sum over α′ ∈ Z^m with α′ ≡ α (mod 2) and h(α′) = h(α) of KhR_0(K(α′);Q) shifted by Σ_i ((w_i − p_i)(α′_i^2 + 2α′_i)/2 − w(K_i(α′_i))) − |α′|, where P is the framing matrix, W the writhe vector, N the crossing matrix (from D), and h(β) = β^T(P−W+N)β.

Background

Building on Theorem 5.5, the authors expect a precise decomposition of S^2_{0,0}(X;α) into contributions from cable Khovanov homologies with specific grading shifts determined by the diagrammatic data (framings, writhes, and crossing matrices). They obtain upper bounds and related structural evidence but note they currently lack a proof of the full isomorphism formula.

A proof would significantly clarify the structure of skein lasagna modules for 2-handlebodies and strengthen diagrammatic nonvanishing results.