Simply-connected manifolds with large homotopy stable classes

For every $k \geq 2$ and $n \geq 2$ we construct $n$ pairwise homotopically inequivalent simply-connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension $4$, we exhibit an analogous phenomenon for spin$^{c}$ structures on $S^2 \times S^2$. For $m\geq 1$, we also provide similar $(4m{-}1)$-connected $8m$-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable $J$-homomorphism $\pi_{4m-1}(SO) \to \pi^s_{4m-1}$.