Strong finiteness of n-generated axial algebras for n > 4

Determine whether every n-generated axial algebra (over the commutative Noetherian base rings considered in the paper) is strongly finite when n > 4; specifically, ascertain whether there exists an integer m such that A = A^{(m)}, the C-submodule spanned by all words in the generators of length at most m, for all n-generated axial algebras with n > 4.

Background

The paper defines A^{(m)} as the C-submodule spanned by all words of length at most m (with arbitrary parenthesization) in the generators, and calls an algebra strongly finite if A = A^{(m)} for some m. It records that for n ≤ 4 this property is known: n = 2 implies A = A^{(2)} (via HRS2), n = 3 implies A = A^{(3)} (via GS), and n = 4 implies A = A^{(7)} (via DRS).

Understanding whether strong finiteness extends to n > 4 affects the structural theory and the construction of generic objects in the axial algebra framework, as many arguments and universal constructions depend on finite spanning by bounded-length words.