Prime labelings on a 3xn grid graph

It is conjectured that the mxn grid graph has a prime labeling for all positive integers m and n. It is known that for any prime p and any integer n such that 1≤n≤p2, there exists a prime labeling on the pxn grid graph Pm x Pn. Also, it is known that the ladder P2 x Pn has a prime labeling for all positive integers n. We assume that Goldbach's Even Conjecture and a strengthened variant of Lemoine's Conjecture are true in order to show that the 3xn grid graph P3 x Pn has a prime labeling for every positive integer n. As a result, P3 x Pn has a prime labeling for every positive integer n≤107.