If I have two finite simplicial 4-complexes embedded linearly in $\mathbb{R}^4$ (as in all the lines and faces are straight and flat and there are only a finite number of 4-simplices) do they have a common refinement/extension (a finite simplicial 4-complex such that every n-simplex in either of the original complexes is contained in a union of n-simplices in the new complex)? Specifically I'm concerned with the case where the two complexes have overlap that isn't 'nice,' i.e. the overlap is not a subcomplex of either complex. Does it help if the complexes are homogenous or convex?
I put 'Hauptvermutung' in quotes because this isn't excatly a question about triangulations; the complexes don't cover the whole space. I feel like this is obviously true but the failure of the Hauptvermutung in $D \geq 5$ and the uncertain status of it in $D=4$ has me spooked. Is this trivial because I've essentially put the two complexes in the same PL structure of $\mathbb{R}^4$?
Edit: I think I've proven that the answer is yes (at least in the homogenous case, although I don't really see an obstruction to the non-homogenous case, since I think you could always refine a non-homogenous simplicial complex to a homogenous simplicial complex in this context). Perhaps someone can tell me if this proof is correct:
Construct the convex hull of both simplicial complexes. Since the simplicial complexes are contained in polytopes, the convex hull will also be a polytope. Extend every face (3-simplex) in both complexes to the intersection of its full hyperplane with the the convex hull (i.e. extend it until it hits the walls of the hull). Now every region bounded by these faces will be a convex polytope (since a compact region constructed from a finite set of bounding hyperplanes is always a convex polytope) and every 4-simplex in either of the original complexes will be the union of some set of these convex polytopes. Now construct the barycentric subdivision of each polytope (which exists because they are convex). Since the barycentric subdivision of a polytope considered along a face is the barycentric subdivision of that face, the new n-simplices (n = 0,1,2) added in this process will match across faces (3-simplices) of adjacent polytopes and the final structure is thus a simplicial complex that refines both original simplicial complexes.