Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms)

Weak form: for the derived $p$-completion $X=S_p^\wedge$, the canonical map $L_{n-1}X\to L_{n-1}L_{K(n)}X$ has no homotopy retraction at height $n=p$ for every $p\ge5$ and at $n=p+1$ for every $p\ge7$; the class $\beta_1^{(p-1)^2}$ is a nonzero kernel element (at $p=5$, $\beta_1^{16}$ in degree 608). Strong form: for $p\ge5$ the map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ is nonzero on $\pi_{-3}$, so the height-three splitting fails even as an equivalence of $E(2)$-local spectra. Companions: at $n=p=3$ the overlap is not in the thick subcategory generated by $L_0S,L_1S,L_2S$; positively, for $p>n+1$ the overlap has a $2^n$-stage filtration with the predicted cofibers (nonsplit), with explicit attachments at height three. Heights below $p-1$ remain open.

Weak form: for the derived $p$-completion $X=S_p^\wedge$, the canonical map $L_{n-1}X\to L_{n-1}L_{K(n)}X$ has no homotopy retraction at height $n=p$ for every $p\ge5$ and at $n=p+1$ for every $p\ge7$; the class $\beta_1^{(p-1)^2}$ is a nonzero kernel element (at $p=5$, $\beta_1^{16}$ in degree 608). Strong form: for $p\ge5$ the map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ is nonzero on $\pi_{-3}$, so the height-three splitting fails even as an equivalence of $E(2)$-local spectra. Companions: at $n=p=3$ the overlap is not in the thick subcategory generated by $L_0S,L_1S,L_2S$; positively, for $p>n+1$ the overlap has a $2^n$-stage filtration with the predicted cofibers (nonsplit), with explicit attachments at height three. Heights below $p-1$ remain open.