Positive matching decomposition number of toroidal grid graphs

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

Background

Theorem 4.5 establishes 5≤pmd(Cm□Cn)≤6 and proves equality to 5 when m+n is odd outside three exceptional pairs. Proposition 4.6 proves that the products C3□C3 and C4□C4 have pmd equal to 6.

The final conjecture identifies the remaining parity and exceptional cases expected to attain the upper bound 6.