Lower bound for Cartesian products up to an additive constant

Establish whether there exists a constant c>0 such that pmd(Γ1)+pmd(Γ2)≤pmd(Γ1□Γ2)+c for all graphs Γ1 and Γ2.

Background

The paper develops several upper bounds for the pmd of Cartesian products but reports no significant general lower bound beyond the maximum valency. The conjecture proposes that the pmd of a Cartesian product cannot be substantially smaller than the sum of the pmds of its factors, except for a universal additive constant.

The authors note that the constant must be positive, citing a later theorem as evidence that an additive-zero bound cannot hold universally.