Suppose $K$ is a compact polytope and $T$ is a piecewise-linear homeomorphism from $K$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that $T^n(x)=x$ for all $x$ in $K$). Can we conclude that a nonperiodic point exists (that is, for some $x$ in $K$ it is the case that $T^n(x) \neq x$ for all $n \geq 1$)?

Note that it need not be the case that for each $n \geq 1$ the set of $x$ in $K$ with $T^n(x) \neq x$ is dense. So my question is not just an application of the Baire category theorem (at least under the most simpleminded approach).

Ideally I would like to draw stronger conclusions from the stated hypotheses (e.g. that the set of nonperiodic points has positive $d$-dimensional measure where $d$ is the dimension of $K$). Some of the hypotheses on $K$ and $T$ are likely to be red herrings, but I include them since they apply for the class of examples that motivated this question.

I have posted a variant of this question in which the piecewise-linearity constraint is dropped; see Nonperiodic points of homeomorphisms of a ball. Montgomery's theorem solves that problem and therefore solves this one, though I'm still hoping for (and will award a bounty to) the best self- contained, clear, and complete proof for the piecewise-linear case.