The real groups construction conjecture for the tame local Langlands correspondence for PGSp(4)

Let $G(F)$ be the group under consideration, let $T(F)$ be the relevant torus, and let ${}^w\Theta_{\chi}$ be the character formula associated with a character $\chi$ and an element $w\in\overline{\overline{W}}$, where $\overline{\overline{W}}=N(G(E),T(F))/T(E)$. Let $Z(F)$ denote the center of $G(F)$, and let $T(F)_{0,s}$ denote the set of strongly regular topologically semisimple elements of $T(F)$. The real groups construction conjecture. For every $w\in\overline{\overline{W}}$, ${}^w\Theta_{\chi}$ agrees on $Z(F)T(F)_{0,s}$ with the character of a unique supercuspidal representation ${}^w\pi$ of $G(F)$. Moreover, the assignment

$$\phi \mapsto \\{{}^w\pi:w\in\overline{\overline{W}}\\}$$

is the local Langlands correspondence for the pure inner forms of $G(F)$. This proposes that the character formula constructed from the tame parameter realizes the local Langlands correspondence through a packet of supercuspidal representations, although the supplied text does not establish the conjecture or provide evidence resolving its status.