Difference in positive matching decomposition under Cartesian product with K2

Determine whether there exists a closed formula for the difference between pmd(Γ□K2) and pmd(Γ) for every graph Γ.

Background

The paper studies positive matching decompositions (pmds) of Cartesian products of graphs. Proposition 2.1 gives general upper bounds in terms of the pmds and chromatic numbers of the factors, but the authors note that these bounds are not always attained. The example pmd(Pm□K2)=3, despite an upper bound of 4, motivates asking whether the change in pmd caused by taking the Cartesian product with K2 admits a closed-form description.