Counterexamples to weak chromatic splitting: sphere kernels and descent exponents

For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height $p+1$ for every prime p ≥ 7. The classical sphere product ${\beta }_1^{(p-1{)}^2}$ gives a nonzero kernel class in both ranges.