If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?

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.

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Gerg, I guess we just use two different terms to define the same notion. I agree with you about dimension 7, there is only one PL structure on $S^7$. But dimension $\ge 7$ are different from $<7$, namely, up to dimension 6 every PL manifold admits a unique smooth structure. In particular all these exotic smooth 4-dimensional manifolds have have exotic PL structures. –  Dmitri Mar 26 '12 at 23:04