Stable Complex Structures On Real Vector Bundles Over Connected Sums Of Quaternionic Projective Spaces

For integers $n\ge 2$ and integers $\ell,m\ge 0$, not both zero, let $M^{4n}_{\ell,m}=\ell\,\mathbb{HP}^{n}\,\#\,m\,\overline{\mathbb{HP}^{n}}$ denote the connected sum of $\ell$ copies of the quaternionic projective space $\mathbb{HP}^{n}$ and $m$ copies of $\mathbb{HP}^{n}$ endowed with the opposi