On compacta not admitting a stable intersection in R^n

Compacta X and Y are said to admit a stable intersection in R^n if there are maps f : X -> R^n and g : Y -> R^n such that for every sufficiently close continuous approximations f' : X -> R^n and g' : Y -> R^n of f and g we have f'(X)\cap g'(Y)\neq\emptyset. The well-known conjecture asserting that X and Y do not admit a stable intersection in R^n if and only if dim X \times Y \leq n-1 was confirmed in many cases. In this paper we prove this conjecture in all the remaining cases except the case dim X =dim Y =3, dim X \times Y=4 and n=5 which still remains open.