A rational obstruction to strong chromatic splitting at height three

For every prime p ≥ 5, the canonical map $L_0{L}_{K(3)}S\to L_0{L}_{K(2)}{L}_{K(3)}S$ for the sphere spectrum S is nonzero on π${}_{-3}$. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying $E(2)$-local spectra without specified summand maps.