Is the misère quotient of Dawson's Kayles (octal $0.07$) infinite at heap size 34? (Games of No Chance A15)

Under the misère convention (the player making the last move LOSES), the misère quotient of an impartial game is the commutative monoid $\mathcal{Q}=\mathcal{A}/{\equiv}$ of positions modulo indistinguishability. Dawson's Kayles is the octal game $0.07$; its misère quotient is finite of order 638 when restricted to positions built from heaps of size $\le 33$. Is the misère quotient infinite once heaps of size $34$ are allowed? If infinite, then by Rédei's theorem it is a finitely presented commutative monoid — exhibit an explicit presentation of $\mathcal{Q}_{34}$; if finite, give its order and a presentation.

Acceptance. FULLY RESOLVES: determine whether the misère quotient of $0.07$ restricted to heaps $\le 34$ is finite or infinite, with a machine-checkable certificate — if finite, its order plus a monoid presentation whose relations are verified against the computed position equivalences; if infinite, a finite presentation of the limiting monoid together with a verification that it does not collapse to a finite quotient. ADVANCES: extend the computed quotient beyond heap 33 with reproducible data, or supply an algorithm (with correctness argument) deciding whether the misère quotient of a given octal game is infinite. Provide the quotient data and a re-derivation script. This is a finite-data target for each fixed heap bound; partial progress (extended, verified quotient data) is itself the product.