  Let $\pi:X\rightarrow\mathbb{P}^n$ be a (holomorphic) Lagrangian fibration
that is very general in the moduli space of Lagrangian fibrations. We
conjecture that the singular fibres in codimension one must be semistable
degenerations of abelian varieties. We prove a partial result towards this
conjecture, and describe an example that provides further evidence.
