Quasi-periodic generating-function conjecture for Sprague–Grundy columns

Consider the array of Sprague–Grundy values for the single-quadrant queens-in-exile problem, and let $x$ be the generating-function variable for a fixed column. Column generating-function conjecture. Every column of the array of Sprague–Grundy values for the single-quadrant problem has a generating function with denominator $(1-x)(1-x^{16})$. The first column is given explicitly in the surrounding discussion, and subsequent columns are observed computationally to have this denominator; the conjecture asserts this pattern for every column.