  In his unpublished notes on fat bundles, W. Ziller poses a compelling
question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with
$\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that
the $G$-orbits are totally geodesic, can one modify $h$ to render all vertical
curvatures equal to $1$? In this note, we establish a rigidity result for fat
Riemannian foliations with bounded holonomy and a specific curvature
constraint. Our result addresses Ziller's question for fat fiber bundles with
compact structure groups, considering connected compact total spaces under a
curvature constraint that holds on various examples, such as locally symmetric
spaces. Additionally, we assume that all vertizontal curvatures coincide at a
point.
