Finiteness of the single-theater misère quotient

Let the single-theater game have escalation set $S=\{1,\dots,m\}$. Its misère quotient is the commutative monoid obtained by identifying positions that are interchangeable in every disjunctive sum, with outcomes recorded by the subset of classes that are losing for the mover. Finiteness conjecture. For every $m$, the misère quotient of the single-theater game is finite. Consequently, multi-theater misère stability is decidable from the image of the joint state in this finite commutative monoid.

The single-theater misère game itself is solved exactly: its $\mathcal{P}$-positions are those with $r\equiv 1\pmod{m+1}$. The conjecture would provide a finite algebraic invariant for deciding multi-theater outcomes, extending the role of Grundy values in normal play; the supplied text gives no resolution status.