In 1996, Muthusamy and Paulraja conjectured that for k ≥ 3, the Cartesian product Km□Kn has a Pk-factorization if and only if mn ≡ 0 mod k and 2(k − 1)|k(m + n − 2). Recently, Chitra and Muthusamy have partially settled this conjecture for k = 3. In this paper, it is shown that for k = 4 the above conjecture is true if (m mod 12, n mod 12) ∈ {(0, 2), (2, 0), (0, 8), (8, 0), (2, 6), (6, 2), (6, 8), (8, 6), (4, 4)}. The left over cases for k = 4 are (m mod 12, n mod 12) ∈ {(0, 5), (5, 0), (0, 11), (11, 0), (1, 4), (4, 1), (3, 8), (8, 3), (4, 7), (7, 4), (4, 10), (10, 4), (8, 9), (9, 8), (10, 10)}.