On closed sets with convex projections under somewhere dense sets of directions

Let $k,n\in\mathbb{N}$ with $k<n$ and let ${\mathcal G}_k(\mathbb{R}^n)$ denote the Grassmann manifold consisting of all $k$-dimensional linear subspaces in $\mathbb{R}^n$. In an earlier paper the authors showed that if the projections of a nonconvex closed set $C\subset\mathbb{R}^n$ are convex and proper for projection directions from some nonempty open set $\mathcal{P}\subset{\mathcal G}_{k}(\mathbb{R}^n)$, then $C$ contains a closed copy of an $(n-k-1)$-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that $\mathcal{P}$ is somewhere dense in ${\mathcal G}_k(\mathbb{R}^n)$.