Call a group presentation $\langle X \,\|\,R \rangle$ minimal if no relator from $R$ is a consequence of the remaining relators, i.e., no $r \in R$ belongs to the normal closure of $R\setminus \{r\}$ in the free group $F(X)$.
Question: Does every finitely generated group have a minimal presentation (with $X$ finite)?
Remark: evidently every finite presentation $\langle X \,\|\,R \rangle$ has a minimal sub-presentation $\langle X \,\|\,R' \rangle$ (where $R' \subseteq R$ has the same normal closure in $F(X)$ as $R$), so the question really concerns infinitely presented groups.
It is possible to construct infinite presentations which do not have minimal sub-presentations. Indeed, let $F=F(a,b)$ be the free group on $\{a,b\}$. One can choose "sufficiently independent" elements $w_1,w_2,\dots \in F$ so that the presentation $$\langle a,b \,\|\, w_i^2,w^2_{i+1}w_i, ~i \in \mathbb{N} \rangle$$ has no minimal sub-presentation. Here $R$ consists of the elements $w_1^2,w_2^2,\dots$, and $w_2^2w_1,w_3^2w_2,\dots$.
However, the above group also has the presentation $\langle a,b \,\|\,w_1,w_2,\dots \rangle$, which may be minimal.
