$\DeclareMathOperator\ncl{ncl}$When I attended a geometric group theory summer school, a question asked by the speaker reminded me of an old [question][1] but shaped in a different manner:
>Given two nontrivial groups $A,B$ and $w \in A \ast B$. For any $k\geq 2$, is the quotient
$(A\ast B)/\ncl(w^k)$ nontrivial, where $\ncl(w^k)$ is the normal closure of $w^k$?

This is true if $B = \mathbb{Z}$ and the projection of $w$ to $B =\mathbb{Z}$ is $\pm 1$. For $k\geq 4$, this was handled by a theorem of Howie.

It is not true in general when $k =1$. For example, let $A = \mathbb{Z}_2$ and $B = \mathbb{Z}_3$
and $w = ab^{-1}$. When $w$ becomes a relation, $a,b$ must be conjugate, but $a,b$ originally have orders $2$ and $3$, resp. It follows that $a = b$ are trivial elements and the group is trivial. I'm particularly interested in the following question
>When $k=1$, are there counterexamples if $A,B$ are torsion-free?


  [1]: https://mathoverflow.net/questions/171776/two-relator-products-of-cyclic-groups