Placing numbers $1,2,\ldots,n^2$ in a square so that numbers of any two adjacent unit subcube are coprime

Question

It is proved HERE that there is a natural number $N$ such that for any $n > N$ it is possible to place numbers $1,2,\cdots, n^2$ is an $n\times n$ square such that the numbers in any two adjacent square (having a mutual edge) are coprime. How can one estimate $N$ !? My guess is that the least such $N$ is equal to $1$.