Two-pile periodicity conjecture for cumulative subtraction games

Consider cumulative subtraction on two piles, where on each turn the active player chooses a pile and plays the single-pile game on that pile. Two-pile periodicity conjecture. The outcome is eventually periodic on any horizontal or vertical line, with period at most

$$2\max S.$$

Eventual periodicity on horizontal and vertical lines is described as known by similar arguments to classical subtraction games; the conjecture strengthens this observation by proposing the explicit period bound. The supplied text gives no evidence of a proof or disproof, so its resolution remains open.