Let $\Gamma$ be a finite connected bipartite graph with colour classes $U$ and $V$ such that:
- every vertex of $U$ has degree $n$, and $n\ge 3$;
- every vertex of $V$ has degree at least $4$;
- $\Gamma$ has diameter $4$;
- for every vertex $u$ of $U$, there are at least $(n+1)/2$ vertices in the neighbourhood $\Gamma(u)$ of $u$ that have degree the minimum degree of $\Gamma(u)$.
Is $\Gamma$ biregular (i.e., the vertices of $V$ have constant degree)? If not, then under what natural and weak conditions is biregularity forced to hold?