TITLE: Local complexity of triangulations
QUESTION [13 upvotes]: According to the introduction in
Cooper, D.; Thurston, W. P., Triangulating 3-manifolds using 5 vertex link types, Topology 27, No. 1, 23-25 (1988). ZBL0656.57004.

It is known that, for any dimension $n$, there is a finite set of link types such that every $n$-manifold has a triangulation in which the link of each vertex is in this set.

(I assume the statement is about PL manifolds.)
What is a proof of or a reference to this known result?
Edit. There is a related discussion by Florian Frick here, but he only gets a bound on the valence of "ridges" (codimension 2 faces). I do not see how to generalize his sketch to faces of higher codimension.  (Actually, I cannot follow his sketch.)

REPLY [2 votes]: Theorem 3.37 of Florian Frick's thesis gives a sketch of the proof of statement in the question for smooth manifolds. At first sight, it only uses the Whitney embedding theorem and transversality, both of which are available for PL manifolds (since you get to pick the embedding you may assume it has a normal microbundle).