Let $F$ be a free group of finite rank $n > 2$, and let $S \subseteq F$ be a subset of cardinality at most $n-2$. Denote by $S^F$ the normal subgroup of $F$ generated by $S$. Must $F/S^F$ be nonabelian?

What if $F$ is free profinite instead? Is it immediate that $F/S^F$ is nontrivial?