If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?
In particular, is this true in $R^{4}$? In 2 and 3 dimensions any two polyhedra
that are homeomorphic are PL-homeomorphic, by theorems of Rado and Moise. In dimension $\geq 5$, this is a trivial special case of theorem 1.1 in
M.A. Armstrong "The Hauptvermutung According to Lashof and Rothenberg" in
*The Hauptvermutung Book*. But I have not found a statement that covers it for dimension 4; and I am not confident that dimension 4 can easily be reduced to
dimension 5.

Also, if anyone can suggest a reference for this particular case that does not go through these very high-powered, difficult, general theorems, I would be interested on stylistic grounds.