Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$.
Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

>What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=e$ for any $i$?

**Postscript.**
The question seems to be open, see [Picture-Hanging Puzzles][1] by Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu.


  [1]: https://arxiv.org/abs/1203.3602