The losing-hook stack conjecture

Let $\lambda$ be a losing partition and let $\Gamma_{r,c}$ be a losing hook. Write

$$\lambda^*=\Gamma_{r,c}\lhd\lambda$$

when the stack operation $\lhd$ is defined. Losing-hook stack conjecture. If $\lambda^*=\Gamma_{r,c}\lhd\lambda$ is defined, then $\lambda^*$ is a $\mathcal P$-position. The conjecture would imply the stated proposition about stacked hooks and motivate further consequences for stacks of odd hooks; it remains open in the source.