Diagonal periodicity conjecture for two-pile cumulative subtraction games

Consider cumulative subtraction on two piles, and fix an integer $k$. A $k$-diagonal is a diagonal half-line of positions of the form $(x,k+x)$. Diagonal periodicity conjecture. The outcome is eventually periodic along any $k$-diagonal. The paper presents this as an observed regularity for two-pile games, but the supplied text gives no evidence of a proof or disproof, so its resolution remains open.