Vector bundles over iterated suspensions of stunted real projective spaces

Let $X_{m,n}^k={\mathrm{\Sigma }}^k(\mathbb{R}{\mathbb{P}}^m\mathrm{/}\mathbb{R}{\mathbb{P}}^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(\xi )=1$ for any vector bundle $\xi$ over $X_{m,n}^k$.