The Chevalley involution conjecture for Langlands-Vogan parameters

Let $G$ be a reductive algebraic group over a local field $F$ that is a pure innerform of a quasi-split group $G_0$ over $F$, equipped with a fixed triple $(B_0,N_0,\psi_0)$ used to parametrize representations on all pure innerforms of $G_0$. Let $\pi$ be an irreducible admissible representation of $G(F)$ with Langlands–Vogan parameter $(\varphi,\mu)$, and let $c_{\widehat{G}}$ denote the Chevalley involution of ${}^LG(\mathbb{C})$. The Chevalley involution conjecture. The Langlands–Vogan parameter of $\pi^\vee$ is

$$\left(c_{\widehat{G}}\circ\varphi,\,(c_{\widehat{G}}\circ\mu)^\vee\otimes\eta_{-1}\right).$$

Here $\eta_{-1}$ is the character of the component group associated with the element acting by $-1$ on the simple root spaces. Since $c_{\widehat{G}}$ acts by $z\mapsto z^{-1}$ on $Z(\widehat{G})$, the resulting character defines the same pure innerform as $\mu$.