The non-finite generation conjecture for identities of mutation algebras

Let $\mathcal{K}(n)$ denote the kernel of the expansion map in degree $n$, and let

$$\mathcal{K}=\bigoplus_{n\geq 1}\mathcal{K}(n)$$

be the resulting operad ideal. Mutation algebras are associative algebras equipped with the mutation operation studied in the paper.

Non-finite generation conjecture. The operad ideal $\mathcal{K}$ is not finitely generated. Equivalently, no finite set of polynomial identities generates all identities satisfied by all mutation algebras.

The claim asserts that genuinely new identities continue to occur in arbitrarily high degrees, rather than all identities following from finitely many lower-degree identities. The computations preceding this statement exhibit large spaces of new identities in degrees $5$ and $6$, but do not establish non-finite generation.