Goodwillie–Whitehead conjecture: strong form

For each $k$, let

$$\delta_k:\Omega^\infty\Sigma L(k)\longrightarrow\Omega^\infty\Sigma L(k+1)$$

be the map in Kuhn's Kahn–Priddy sequence, and let

$$\delta^{S^1}_k:\Omega^\infty\Sigma^{1-k}L(k)_1\longrightarrow\Omega^\infty\Sigma^{1-k}L(k+1)_1$$

be the $k$-invariant map of the Goodwillie tower for $S^1$. Goodwillie–Whitehead conjecture, strong form.

$$\Omega^k\delta_k=\delta^{S^1}_k.$$

The theorem in the source establishes the $k=0$ case, while the full compatibility is presented as an open calculus version of the Whitehead conjecture. Kuhn is described as proposing an even stronger natural-transformation-level version.