Positive matching decomposition formula for Cartesian products of trees

Prove or disprove that, for trees T1,…,Tn with at most three factors isomorphic to K2, the equality pmd(T1□⋯□Tn)=Δ(T1□⋯□Tn)+m−δm≠0 holds, where m is the number of factors isomorphic to K2.

Background

Corollary 2.4 establishes the proposed equality for several cases, including products of trees without K2 factors and products with one K2 factor. The conjecture extends these exact formulas to products of trees with m≤3 K2 factors. The subsequent remark states that the analogous upper bound fails when m≥4, so the conjecture is explicitly restricted to at most three such factors.