Non-kinetic bundles from nontrivial smooth SU(2)-actions

Let $W$ be an arbitrary closed manifold on which $SU(2)$ acts smoothly and non-trivially. Let $s\colon BSU(2) \to B\operatorname{Diff}(W)$ be the induced map on classifying spaces, and let $\psi_k\colon BSU(2)\to BSU(2)$ be the map appearing in the conjecture. For an odd integer $k>1$, the bundle classified by

$$s\circ\psi_k\colon BSU(2)\to B\operatorname{Diff}(W)$$

is non-kinetic. Non-kinetic bundle conjecture. For every odd integer $k>1$, the bundle classified by $s\circ\psi_k$ is non-kinetic. This conjecture proposes that nontrivial smooth $SU(2)$-actions produce genuinely non-kinetic bundles after applying the specified self-map of $BSU(2)$; its status is not resolved in the supplied text.