Flammenkamp's conjecture on exponential periods in subtraction games

Let a subtraction game be played on the non-negative integers using a finite ruleset $S\subset\mathbb{N}=\{1,2,\ldots\}$. The current player subtracts an element of $S$ when possible, and the player unable to move loses. The eventual period length is the period of the outcome sequence after a finite preperiod. Flammenkamp's conjecture. There exists a sequence of subtraction games on finite rulesets with an exponential eventual period length of the outcomes, with respect to the largest member of the subtraction set. This challenges the classical expectation that period lengths are bounded polynomially in the largest subtraction.