The bu Whitehead conjecture

Let $A_m$ denote the spectrum associated to finite-dimensional complex vector spaces and unitary isomorphisms, and work at a prime $p$. Define

$$T(k)=\Sigma^{-(k+1)}A_{p^k}/A_{p^k-1}.$$

The resulting chain complex is

$$\dots\longrightarrow T(2)\longrightarrow T(1)\longrightarrow T(0)\longrightarrow bu\longrightarrow \operatorname{H}{\mathbb{Z}}.$$

$bu$ Whitehead conjecture. The complex is exact at the prime $p$.

This is proposed as a $bu$-analogue of the classical Whitehead conjecture, which was proved by Kuhn and Priddy. The paper gives supporting evidence but does not establish exactness.