The non-indexedness conjecture for the word problem of $\mathbb Z^2$

Let $\mathbb Z^2$ be the free abelian group of rank $2$, and let its word problem be the language of words over a finite generating set that represent the identity element of $\mathbb Z^2$. Non-indexedness conjecture. The word problem for $\mathbb Z^2$ is not indexed. The paper places this question in the Chomsky hierarchy: word problems of finitely generated nilpotent groups are context-sensitive, while it remains unknown whether the word problem of $\mathbb Z^n$ is accepted by a nested stack automaton without extra restrictions.