Exact value for Cartesian products of two cycles

Prove that pmd(Cm □ Cn) = 6 for all integers 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 for all m, n ≥ 3 and proves that the value is 5 when m + n is odd, apart from the exceptional pairs (3, 4), (3, 6), and (5, 6). Proposition 4.6 proves value 6 for (3, 3) and (4, 4). The conjecture asserts that all remaining even-parity cases and the three exceptional odd-parity cases attain the upper bound 6.