Additive lower-bound conjecture for positive matching decomposition numbers

Let $\Gamma_1$ and $\Gamma_2$ be graphs, let $\Gamma_1\square\Gamma_2$ denote their Cartesian product, and let $\mathrm{pmd}(\Gamma)$ denote the positive matching decomposition number. Additive lower-bound conjecture for positive matching decomposition numbers. There exists a constant $c>0$ such that

$$\mathrm{pmd}(\Gamma_1)+\mathrm{pmd}(\Gamma_2)\leq\mathrm{pmd}(\Gamma_1\square\Gamma_2)+c$$

for all graphs $\Gamma_1$ and $\Gamma_2$. The paper notes that its results primarily give upper bounds, while the available general lower bound is only through maximum valency. This conjecture proposes that the product's positive matching decomposition number is bounded below by the sum of those of its factors, up to an additive universal constant.