The Hahn-Wilson conjecture: fp-type at most n spectra are exactly the thick subcategory generated by BP<n>

Claims that for every sufficiently large prime $p$ there is a connective $p$-complete spectrum $X$ of exact fp-type two with $X\notin\mathrm{Thick}(\mathrm{BP}\langle2\rangle_p^\wedge)$ for a standard form of $\mathrm{BP}\langle2\rangle$, refuting the universally quantified conjecture at height two. The same $X$ satisfies $L_2^fX\simeq L_2X$ and $L_{T(2)}X\simeq L_{K(2)}X$, so the obstruction lies in how the chromatic pieces are glued, not in a telescope-type failure; the monochromatic form proved by Lee-Pstragowski is untouched. It does NOT treat small primes, heights above two, or the height-one case, which Lee-Pstragowski proved.

Claims that for every sufficiently large prime $p$ there is a connective $p$-complete spectrum $X$ of exact fp-type two with $X\notin\mathrm{Thick}(\mathrm{BP}\langle2\rangle_p^\wedge)$ for a standard form of $\mathrm{BP}\langle2\rangle$, refuting the universally quantified conjecture at height two. The same $X$ satisfies $L_2^fX\simeq L_2X$ and $L_{T(2)}X\simeq L_{K(2)}X$, so the obstruction lies in how the chromatic pieces are glued, not in a telescope-type failure; the monochromatic form proved by Lee-Pstragowski is untouched. It does NOT treat small primes, heights above two, or the height-one case, which Lee-Pstragowski proved.