Vogan's geometric form of the local Langlands correspondence

Let $\check G$ be a connected reductive algebraic group over $F=\mathbb{Q}_p$, let $G$ be its Langlands dual group, and let $\Lambda:W_F\to{}^L\check G$ be an infinitesimal character. Let $E_\Lambda$ be the Vogan variety with its $G_\Lambda=Z_G(\Lambda)$-action, and let $\operatorname{Rep}_\Lambda(\check G_\delta)$ denote the finite-length category of smooth representations of the pure inner form $\check G_\delta$ whose irreducible factors have infinitesimal character $\Lambda$. Vogan's local Langlands correspondence. There is a perfect pairing

$$\Big(\bigoplus_{\delta\in H^1(F,\check G)}K\operatorname{Rep}_\Lambda(\check G_\delta)\Big)\times K D^b(E_\Lambda,G_\Lambda)\to\mathbb{Z}$$

such that the classes of simple, respectively standard, objects on both sides form dual bases up to some signs. In particular, the change-of-basis matrices on both sides between simple objects and standard objects are transposes of each other. This is a geometric formulation of the local Langlands correspondence relating representation categories of pure inner forms to equivariant sheaves on the Vogan variety. The excerpt gives no resolution of the assertion, so it remains open.