The L-packets conjecture for Bernstein components

Let $F$ be a non-archimedean local field, $G$ a connected reductive algebraic group over $F$, and ${\mathfrak s}=[L,\omega]_G$ an inertial equivalence class. Let $T_{\mathfrak s}$ be the Bernstein torus, $W_{\mathfrak s}$ its finite stabilizer, and suppose a bijection $\operatorname{Irr}^{\mathfrak s}(G)\leftrightarrow (T_{\mathfrak s}/\!/W_{\mathfrak s})_{\natural}$ as above. For $t\in T_{\mathfrak s}$, let $W_{{\mathfrak s},t}$ be its isotropy group, with decomposition $W_{{\mathfrak s},t}=W(R_{{\mathfrak s},t})\rtimes {\mathfrak R}_{{\mathfrak s},t}$, where $R_{{\mathfrak s},t}$ is the associated root system. L-packets. Assume that a local Langlands correspondence exists for $\operatorname{Irr}^{\mathfrak s}(G)$. Then the twisting cocycle factors through $W_{{\mathfrak s},t}/W(R_{{\mathfrak s},t})$, the bijection is canonical up to permutations within L-packets, and two representations with images $[t,\rho]$ and $[t',\rho']$ lie in the same L-packet exactly when there is $w\in W_{\mathfrak s}$ with $wt'=t$ and the corresponding Springer parameters have the same unipotent class. The conjecture refines the extended-quotient parametrization by predicting how its parameters organize into L-packets; the supplied source gives no resolution status.