A very natural solid here is a pentagonal trapezohedron: it has 10 four-sided faces. Each face is a kite-shaped quadrilateral. This version uses a gappy wireframe with thinner rear lines.
This is a clean visual model rather than a proof from the earlier 14-point planar layout. The hint “there are now 10 four-sided shapes” strongly suggests a 10-faced quadrilateral solid.