Tiling by incongruent equilateral triangles without requiring local finiteness

It has been shown by Tutte [7] that an equilateral triangle cannot be dissected into n equilateral triangles of pairwise different sizes, n ∈ {2, 3, . . .}. A dissection into equilateral triangles of that kind is even impossible for arbitrary convex polygons (see [2, 8]). Is there a tiling of the Euclidean plane by equilateral triangles all of different sizes? The answer to this question posed in [4, Exercise 2.4.10] and [3, Section C11] depends on what one means by a tiling: Scherer [6] proves that the plane cannot be dissected into incongruent equilateral triangles such that one of them is of minimal size. Tuza [8] notes that countably many equilateral triangles of different sizes can be packed into a given equilateral triangle such that only a remainder of measure zero is left uncovered. In the same way one can tile the whole plane except for a set of measure zero. Klaaßen [5]