Difference between the pmd of a Cartesian product with K2 and the original graph

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

Background

The paper establishes upper bounds for the positive matching decomposition number of Cartesian products in terms of the pmds and chromatic numbers of the factor graphs. The authors note that the bound from Proposition 2.1 is sharp for many graphs, but need not be attained in general; for example, pmd(Pm □ K2) = 3 while the corresponding upper bound is 4.

This motivates the unresolved question of whether the change in pmd caused by taking the Cartesian product with K2 can be expressed by a closed formula depending on Γ.