What is the simplest example of a smooth subdivision of the standard simplex $\Delta^n$ which can not be realized as a rectilinear subdivision?

That is I want a simplicial complex $K$ for which there is a homeomorphism $|K| \to \Delta^n$ that is smooth when restricted to each simplex and such that there is no homeomorphism $|K| \to \Delta^n$ which is linear when restricted to each simplex.

If one replaces "smooth" with "continuous", there are plenty of examples available, but for all of them (that I've seen) there is not even a piecewise linear homeomorphism $|K| \to \Delta^n$. Since that's impossible for smooth subdivisions such examples are clearly not smoothable.


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