Conjecture on the stable connectivity of the covariant part of the $A$-theory transformation

Let $\theta=(B,p,xi)$ be a family of $Rd$-bundles, and let T^% denote the covariant part of the transformation \tau^% associated to the extended $A$-theory characteristic. Here $d$ is the manifold dimension. Stable connectivity conjecture. The connectivity of the covariant part of \tau^% increases to $\infty$ as $d$ increases to $\infty$. This expectation is motivated by analogies with the smooth case, results on the homotopy type of the topological cobordism category, and the assembly map of $A$-theory; the source supplies no resolution.