Replacement conjecture for equivalent impartial misère-game positions

Let $\xi$ be a position with misère outcome $o^-(\xi)=\mathcal{P}$, and suppose its misère quotient is isomorphic to that of the impartial position $*$. In symbols, assume

$$\mathscr{M}_{\mathit{c}\ell(\xi)}\cong\mathscr{M}_{\mathit{c}\ell(*)}.$$

Replacement conjecture. One can replace $*$ by $\xi$ in any position having $*$ as an option without changing the resulting misère monoid. This would provide a substitution principle for positions with the same relevant misère quotient, but the paper presents the claim as conjectural and gives no proof or resolution.