Positive matching decomposition number of Cartesian products of trees

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

Background

Corollary 2.4 proves the proposed formula for Cartesian products of trees in the cases with no K2 factor and with exactly one K2 factor, and gives an upper bound involving the number m of K2 factors. The authors then formulate the displayed equality as a conjecture for m ≤ 3. They subsequently note that the conjecture fails when m ≥ 4, so the unresolved claim is restricted to the stated range.