Closed formula for the Cartesian product with K2

Determine whether there exists a closed formula for the difference between the positive matching decomposition numbers 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 provides general upper bounds for pmd(Γ1□Γ2) in terms of the pmds and chromatic numbers of the factors, but the authors note that these bounds are not always attained.

For the specific product Γ□K2, the paper gives examples where the general upper bound is not sharp, motivating the question of whether the change in pmd caused by taking the Cartesian product with K2 can nevertheless be described by a closed formula.