Is a homotopy sphere with maximum Morse perfection actually diffeomorphic to a standard sphere?

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
$$p:S^ktimesSigma^nrightarrowmathbb{R}$$
where $S^k$ is the standard sphere, such that

(i). For any $xin S^k$, $p|_{(x,Sigma)}$ 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?