Does a homotopy sphere that bounds a highly connected manifold also bound a parallelizable manifold?

Question

Suppose that the homotopy sphere $\Sigma^{n}$ can be realized as the boundary of a smooth $(n+1)$-dimensional cobordism that is $(n-1)/2$-connected for $n$ odd (respectively, $(n-2)/2$-connected for $n$ even). For which values of $n$ does this imply that the element $[\Sigma^{n}] \in \Gamma_{n}$ lies in the subgroup $bP_{n+1} \subset \Gamma_{n}$?