ELMO 2023 Problem 6

For a set $S$ of positive integers and a positive integer $n$, consider the game of $(n,S)$-nim, which is as follows. A pile starts with $n$ watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of $S$. The last player to move wins. Let $f(S)$ denote the set of positive integers $n$ for which Deric has a winning strategy in $(n,S)$-nim.

Let $T$ be a set of positive integers. Must the sequence $$T, f(T), f(f(T)), \;\ldots$$ be eventually constant?

Proposed by Brandon Wang and Edward Wan