Realizing non-suspension co-H-spaces as Conley indices of isolated critical points

Determine whether there exists a smooth function f: R^{m+1} → R with an isolated critical point at the origin such that the Conley index C({0}) of the isolated invariant set {0} (for the negative gradient flow of f) is a co-H-space that is not homotopy equivalent to a suspension.

Background

For a smooth function f on a manifold X, the Conley index C(S) of an isolated invariant set S for the gradient flow is a homotopy invariant capturing local dynamics. When S is an isolated critical point x that is not a minimum, C({x}) is a co-H-space. Pears proved that for any finite CW-complex Z, there exists a smooth function on R^{\tilde{m}+1} with an isolated critical point at the origin whose Conley index is the suspension \Sigma Z, showing that suspensions can occur as such Conley indices.

However, while there are co-H-spaces that are not suspensions, it is unresolved whether any such non-suspension co-H-space can occur as the Conley index of an isolated critical point of a smooth function on Euclidean space. Establishing this would clarify the range of homotopy types realizable as local Conley indices of isolated singularities.