The Morse perfection of a closed differentiable manifold $\Sigma^n$ is defined to be the largest integer $k$, such that there exists a smooth mapping $$p:S^k\times\Sigma^n\rightarrow\mathbb{R}$$ where $S^k$ is the standard sphere, such that
(i). For any $x\in S^k$, $p|_{(x,\Sigma)}$ restricts to a Morse function with two critical points over $\Sigma$.
(ii). $p|_{(x,\Sigma)}=-p|_{(-x,\Sigma)}$.
Clearly, a manifold with positive Morse perfection is a topological sphere. Using Borsuk-Ulam theorem we know that the Morse perfection of a homotopy sphere is no greater than its dimension.
My question is, what happens if a homotopy sphere has Morse perfection equal to its dimension, is it diffeomorphic to the standard sphere?
