The duality cofiber-sequence conjecture for the K(2)-local sphere

Let $p$ be a prime and let $\ell$ be a generator of $\mathbb{Z}_p^\times$ when $p$ is odd, or of $\mathbb{Z}_2^\times/\{\pm 1\}$ when $p=2$. Write $Q(\ell)_{K(2)}$ for the $K(2)$-localization of $Q(\ell)$, let $\eta$ denote the $K(2)$-localized unit map, and let $D_{K(2)}$ denote Spanier–Whitehead duality in the $K(2)$-local category. Duality cofiber-sequence conjecture. If $p$ is odd, then

$$D_{K(2)}Q(\ell)\xrightarrow{D\eta}S_{K(2)}\xrightarrow{\eta}Q(\ell)_{K(2)}$$

is a cofiber sequence. At $p=2$, let $\widetilde{S}=E_2^{h\widetilde{\mathbb{G}}_2}$, where $\widetilde{\mathbb{G}}_2=\widetilde{\mathbb{S}}_2\rtimes Gal$ is an index-$2$ subgroup of $\mathbb{G}_2$, with $\widetilde{\mathbb{S}}_2$ the group specified in the cited theorem. Then

$$D_{\widetilde{S},K(2)}Q(\ell)\xrightarrow{D\eta}\widetilde{S}\xrightarrow{\eta}Q(\ell)_{K(2)}$$

is a cofiber sequence, where $D_{\widetilde{S},K(2)}$ denotes Spanier–Whitehead duality in the category of $K(2)$-local $\widetilde{S}$-modules. This conjecture proposes that the $K(2)$-local sphere is obtained from $Q(\ell)_{K(2)}$ and its appropriate dual by such a cofiber sequence; its resolution would describe a fundamental relationship between the $K(2)$-local sphere and the spectra arising from topological modular forms.