  A double disk bundle is any smooth closed manifold obtained as the union of
the total spaces of two disk bundles, glued together along their common
boundary. The Double Soul Conjecture asserts that a closed simply connected
manifold admitting a metric of non-negative sectional curvature is necessarily
a double disk bundle. We study a generalization of this conjecture by dropping
the requirement that the manifold be simply connected. Previously, a unique
counterexample was known to this generalization, the Poincar\'e dodecahedral
space $S^3/I^\ast$. We find infinitely many $3$-dimensional counterexamples, as
well as another infinite family of flat counterexamples whose dimensions grow
without bound.
