Dimension-5 case of the main semiflow homotopy result

Prove or disprove that Proposition prop:GAS:homotopy:semiflow (which establishes a stability-preserving homotopy of semiflows for n≠5) also holds in dimension n=5.

Background

Proposition prop:GAS:homotopy:semiflow asserts that, for n≠5, if the origin is strongly globally asymptotically stable under the Filippov regularization of a sufficiently regular vector field, then there exists a homotopy of semiflows to the canonical exponential flow that preserves stability.

The dimension n=5 is excluded because the needed diffeomorphism regularity of the conjugating map is not established in that case; resolving n=5 would close this gap.