Sprague–Grundy values of King on odd-by-odd generalized staircases

Determine the Sprague–Grundy value of the King game on the generalized staircase partition consisting of r equal staircase blocks of height r, for odd r and k ≥ 1, according to the three cases specified by the parity of k modulo r + 2.

Background

The paper studies impartial chess games on Young diagrams of integer partitions. In the King game, a move removes a positive number of cells from the first row, first column, or both simultaneously by the same amount. The authors derive several exact Sprague–Grundy formulas for King on rectangles, single-row staircases, and certain generalized staircases.

Conjecture 5.1 concerns the unresolved case of generalized staircases with equal dimensions and odd side parameter r. It predicts that the Sprague–Grundy value depends only on the residue class of k modulo r + 2, taking values 0, 1, or 2 as specified in the conjecture.