On closed sets with convex projections under narrow sets of directions

Dijkstra, Goodsell, and Wright have shown that if a nonconvex compactum in $\mathbb{R}^n$ has the property that its projection onto all $k$-dimensional planes is convex, then the compactum contains a topological copy of the $(k-1)$-sphere. This theorem was extended over the class of unbounded closed sets by Barov, Cobb, and Dijkstra. We show that the results in these two papers remain valid under the much weaker assumption that the collection of projection directions has a nonempty interior.