The even-by-even hook Conway-pair conjecture

A partition is a Young diagram, and its Conway pair is the pair of game-theoretic invariants used in the Conway–Gurvich–Ho classification. An even-by-even hook is a hook with an even number of rows and an even number of columns. Even-by-even hook conjecture. A partition $\lambda$ has Conway pair $(0,1)$ if and only if $\lambda$ is an even-by-even hook. Conway pairs $(0,1)$ and $(1,0)$ play an important role in the classification scheme; the source leaves this characterization as a conjecture.