Positive matching decomposition number of cycle-cycle grids

Prove that pmd(Cm□Cn)=6 for all m,n≥3 such that m+n is even or (m,n)∈{(3,4),(3,6),(5,6)}.

Background

Theorem 4.5 proves that 5≤pmd(Cm□Cn)≤6 for all m,n≥3 and establishes equality to 5 when m+n is odd, apart from the exceptional pairs (3,4), (3,6), and (5,6). Proposition 4.6 proves pmd(C3□C3)=pmd(C4□C4)=6.

The remaining cases are conjectured to attain the upper bound 6: all pairs with even m+n and the three exceptional odd-sum pairs identified in Theorem 4.5.