Winning-position formula for three-pile Nim with a pass-move

Determine a mathematical formula describing the previous player's winning positions in classical three-pile Nim when a pass-move may be used at most once and may not be played from a terminal position.

Background

Classical three-pile Nim without a pass-move has a complete characterization of previous-player winning positions: a position with pile sizes x, y, and z is winning for the previous player exactly when x ⊕ y ⊕ z = 0. The paper studies a modified ruleset in which either player may pass once during the game, after which no further pass is available, and passing from a terminal position is forbidden.

The paper establishes formulas for several classes of chocolate games with a pass-move by exploiting their decomposition into component games. However, it explicitly identifies the corresponding general formula for classical three-pile Nim with a pass-move as unknown; this remains unresolved and is distinct from the special chocolate-game cases treated in the paper.