$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$.

Let $\pi$ be an irreducible generic admissible representation of $G$. Let $\pi^e$ be the $e$-conjugate of $\pi$ defined by $\pi^e(g):= \pi(e^{-1}g e)$.

Let $\sigma$ and $\sigma^e$ be the Weil group representations corresponding to $\pi$ and $\pi^e$ through local Langlands correspondence for $G=\SO(2,0)(\mathbb{R})$.

Then I am wondering whether the Artin gamma functions of $\sigma$ and $\sigma^e$ are equal.