Nonpolynomiality conjecture for magic-square counting functions

Let $M_n$, $S_n$, and $P_n$ denote the counting functions for the magic squares considered in the paper, with parameter $n$. Nonpolynomiality conjecture. The functions $M_n$, $S_n$, and $P_n$ are not polynomials when $n\geq 5$. This conjecture concerns the periods of the counting functions, which are expected to increase with $n$. Results cited in the source support it; the conjecture is presented as open.