We explore connections between our earlier work, in which we constructed
spectra that interpolate between bu and HZ, and earlier work of Kuhn and Priddy
on the Whitehead conjecture and of Rognes on the stable rank filtration in
algebraic K-theory. We construct a "chain complex of spectra" that is a
bu-analogue of an auxiliary complex used by Kuhn-Priddy; we conjecture that
this chain complex is "exact"; and we give some supporting evidence. We tie
this to work of Rognes by showing that our auxiliary complex can be constructed
in terms of the stable rank filtration. As a by-product, we verify for the case
of topological complex K-theory a conjecture made by Rognes about the
connectivity (for certain rings) of the filtration subquotients of the stable
rank filtration of algebraic K-theory.