Positive matching decomposition number of path–cycle grid graphs

Prove that for all integers m, n ≥ 3, the positive matching decomposition number satisfies pmd(Pm □ Cn) = 5 whenever n is even with n < 4(m − 1) or n is odd with n < 2m.

Background

Theorem 4.1 proves that 4 ≤ pmd(Pm □ Cn) ≤ 5 for all m, n ≥ 3 and establishes equality pmd(Pm □ Cn) = 4 in the complementary large-n regimes: even n ≥ 4(m − 1) and odd n > 2m. Direct computations give pmd(P3 □ Cn) = 5 for n = 3, 4, 5, 6, motivating the conjecture that the remaining parameter ranges uniformly have value 5.