Local Langlands conjecture for generic supercuspidal representations of $PGSp_6$

Let $PGSp_6$ be the split adjoint group, and let $\operatorname{Irr}_g^\circ(PGSp_6)$ denote the set of isomorphism classes of its generic supercuspidal irreducible representations. Let $\operatorname{Par}^\circ(PGSp_6)$ denote the corresponding set of parameters, modulo the adjoint action of $Spin_7(\mathbb C)$. The local Langlands conjecture for $PGSp_6$. There is a bijection

$$\Phi(PGSp_6) \colon \operatorname{Irr}_g^\circ(PGSp_6) \rightarrow \frac{\operatorname{Par}^\circ(PGSp_6)}{\operatorname{Ad}(Spin_7(\mathbb C))},$$

in which Shahidi's degree 8 Spin $L$-function on irreducible representations corresponds to the Artin–Weil degree 8 $L$-function associated to the Spin representation of $Spin_7(\mathbb C)$. The local Langlands conjectures are known for $PGL_3$, while corresponding results for $PGSp_6$ were open at the time of the source and were expected to be proved in the near future.