Characterization of simple impartial games

Characterize which families of known impartial games are simple and establish whether Fair Shares and Varied Pairs is simple for any number of almonds.

Background

The paper defines simplicity for optiongraphs through the triviality of the maximum congruence relation and notes that simplicity behaves differently for cyclic optiongraphs and rulegraphs. It asks for families of known games whose associated optiongraphs are simple. The authors explicitly identify Fair Shares and Varied Pairs as a conjectural example, with simplicity asserted for every number of almonds still unresolved.