The local Langlands parametrization conjecture for disconnected even orthogonal groups

Let $H$ be a $p$-adic symplectic group or split special orthogonal group, and in the special even orthogonal case let $H^+$ be the corresponding full even orthogonal group. Let $\Phi(H^+)_{\lambda^{\mathbb{Q}_p}}$ be the quotient of the parameter set by $\widehat{H}^+$-conjugation, and define $\Xi(\lambda^{\mathbb{Q}_p},H^+)$ using the component-group characters $\widehat{A}^+_{\phi^{\mathbb{Q}_p}}$. The local Langlands parametrization conjecture. There should be a bijection

$$\Pi_{{\mathrm{m}\,\mathrm{pure}}}(\lambda^{\mathbb{Q}_p},H^+)\xrightarrow{\sim}\Xi(\lambda^{\mathbb{Q}_p},H^+):=\{(\phi^{\mathbb{Q}_p},\epsilon)\mid \phi^{\mathbb{Q}_p}\in\Phi(H^+)_{\lambda^{\mathbb{Q}_p}},\ \epsilon\in\widehat{A}^+_{\phi^{\mathbb{Q}_p}}\}.$$

This is the expected extension of the local Langlands correspondence from connected reductive groups to the disconnected full even orthogonal group; the source gives no resolution status.