Refined local Langlands correspondence for p-adic quasi-split groups

Let ${^LG}=\widehat{G}\rtimes\Gamma$ be the $L$-group of a quasi-split group $\mathbf G$, and let $\varphi:\mathrm{WD}_F\to {^LG}$ be an $L$-parameter with $L$-packet $\Pi_\varphi$. Define $S_\varphi=Z_{\widehat{G}}(\varphi(\mathrm{WD}_F))$, $\overline{S}_\varphi=S_\varphi/Z(\widehat{G})^\Gamma$, and let $\pi_0(\overline{S}_\varphi)$ be the component group. A Whittaker datum is denoted by $\mathfrak w$.

Refined local Langlands correspondence for $p$-adic quasi-split groups. The following should hold: for every $L$-parameter $\varphi$, there is a bijection

$$\Pi_\varphi(\mathbf G)\longrightarrow \operatorname{Irr}(\pi_0(\overline{S}_\varphi));$$

for every Whittaker datum $\mathfrak w$, each tempered $L$-packet contains a unique $\mathfrak w$-generic representation; for tempered $\varphi$, there is a bijection $\iota_{\mathfrak w}$ as above, depending on $\mathfrak w$, that maps this generic representation to the trivial representation; and the bijection is characterized by suitable character identities, with $\dim(\iota_{\mathfrak w}(\pi))$ independent of $\mathfrak w$.

These are desiderata for the refined local Langlands correspondence and are used as working hypotheses in the paper. The source does not indicate a general resolution.