On the unstable intersection conjecture

Compacta X and Y are said to admit a stable intersection in ℝⁿ if there are maps f: X → ℝⁿ and g: Y → ℝⁿ such that for every sufficiently close continuous approximations f': X → ℝⁿ and g': Y → ℝⁿ of f and g, we have f' (X) ∩ g' (Y) ≠ ∅. The unstable intersection conjecture asserts that X and Y do not admit a stable intersection in ℝⁿ if and only if dim X × Y ≤ n − 1. This conjecture was intensively studied and confirmed in many cases. we prove the unstable intersection conjecture in all the remaining cases except the case dim X = dim Y = 3, dim X × Y = 4 and n = 5, which still remains open.