Counterexamples to the Hahn-Wilson conjecture at height two

We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of $\mathrm{BP}⟨2{⟩}_p^{\wedge }$. The same spectrum satisfies both localization comparisons $L_2^{f}X\simeq L_{2}X$ and $L_{T(2)}X\simeq L_{K(2)}X$.