The conjecture on algebra structures of K(1) at p=2

For $p=2$, let $K(1)$ be the spectrum considered as an algebra over the completed connective complex $K$-theory spectrum $\widehat{KU}_2$, and let an opposite algebra structure mean the structure obtained by reversing multiplication. Algebra-structure conjecture. There are precisely two structures of an $\widehat{KU}_2$-algebra on $K(1)$, and these are opposite to each other. Moreover, the spectrum $K(1)$ possesses a unique structure of an $\mathbb{S}$-algebra. The preceding discussion shows that the two $\widehat{KU}_2$-algebra structures, if they exist, are not isomorphic as $\widehat{KU}_2$-algebras, while $K(1)$ is nevertheless isomorphic to its opposite as a ring spectrum; the source gives no resolution of the conjecture.