ETOG representation conjecture for bi-positional winning conditions

A bi-positional winning condition is one for which both the condition and its complement are half-positional. A winning condition is prefix-independent if membership is unchanged after removing or adding finite prefixes. An energy condition over a totally ordered group is defined from edge colors valued in a totally ordered group, with Alice winning when the sequence of partial sums has an infinite decreasing subsequence. Two infinite sequences are periodic when they are repetitions of a finite non-empty word.

ETOG representation conjecture. Every bi-positional prefix-independent winning condition coincides on periodic sequences with some energy condition over a totally ordered group.

The conjecture proposes that energy conditions over totally ordered groups capture the behavior of all bi-positional prefix-independent winning conditions on periodic sequences. The paper introduces these energy conditions and proves their bi-positionality, but does not resolve the representation claim.