Exact pmd of Cartesian products of cycles

Prove that pmd(Cm□Cn)=6 for all m,n≥3 satisfying m+n even or (m,n)∈{(3,4),(3,6),(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 pmd(C3□C3)=pmd(C4□C4)=6. The remaining cases are conjectured to attain the upper bound 6 precisely when the cycle lengths have even total parity or belong to the three listed exceptional pairs.