An infrasolvmanifold which does not bound

Every 4-dimensional infrasolvmanifold $M$ with ${\beta }_1(M;\mathbb{Q})>0$ or which is flat or has one of the geometries $\mathbb{N}i{l}^4$, $\mathbb{S}o{l}_{m,n}^4$, or $\mathbb{S}o{l}_0^4$ bounds. However there are non-orientable $\mathbb{S}o{l}_1^4$-manifolds which do not bound. The question remains open for $\mathbb{N}i{l}^3\times {\mathbb{E}}^1$-manifolds. We also give simple cobounding 5-manifolds for all but five of the 74 flat 4-manifolds, and investigate which orientable flat 4-manifolds embed in ${\mathbb{R}}^5$.