Are all convex polyhedra fully truncable? (Q2827610)

The author calls a convex polyhedron in ordinary space fully truncable (\textit{sic}) if each of its vertices can by truncated by a plane in such a way that planes truncating adjacent vertices meet in a common point on the edge that joins them. The author shows by example that not all polyhedra have this property. On the other hand, it is known that for each polyhedron there is an isomorphic one whose edges all touch some sphere, which therefore does have the property.