Bott periodicity implies that $\Omega(SU)\simeq G(\infty)$. Here, by $G(\infty)$, I mean the direct limit $\underset{m\to \infty}{\lim} G_m(\mathbb{C}^{2m})$ where $G_m(\mathbb{C}^{2m})\subset G_{m+1}(\mathbb{C}^{2m+2})$ by stabilization (or some similar nice model for $BU$). This is the classifying space for $U = \underset{m\to\infty}{\lim} U(m)$, and may be identified with $U/(U\times U)$, where the coordinates of each factor of the product subgroup alternate. From the path space construction, one knows that there is a contractible Serre fibration $\Omega SU \to E \to SU$, where $E$ is the space of paths in $SU$ from the identity. The fibers are each homotopic to $G(\infty)$.

The question I have is whether this may be realized by a contractible fiber bundle $G(\infty)\to E \to SU$ ? Other quasi-fibrations were given by Aguilar and Prieto, and by Behrens. There are also fibrations of this sort constructed using symplectic reduction by Latour and by Giroux, where the fibers are homotopic to $G(\infty)$. What I'm asking for is whether there exists a contractible fiber bundle rather than just a fibration or quasi-fibration?

Presumably such a bundle would arise from a map $SU \to BDiff(G(\infty))$. The isometry group of $G(\infty)$ contains $SU$, so one could ask *a fortiori* whether there is a map $SU \to BSU$, that is a map $f: SU \to G(\infty)$, such that the induced fiber bundle is contractible? **Added clarification:** The induced bundle would come from the fiber product of the pull-back of the $U$ bundle $U\to E \to G(\infty)$ with the action of $U$ on $G(\infty)$: $f^{*}(E) \times_{U} G(\infty)$.