A spectral expression for a certain orbital integral

Let $F$ be a $p$-adic field, $G = \text{GL}_{2n}(F)$ and $\theta_0$ be the exterior automorphism of $G$ that fixes a pinning of a Borel pair. Consider the set $\widetilde{G} = G \theta_0$ on which $G$ acts by conjugacy and the orbital integral $J_{\widetilde{G}}(\theta_0, f)$ at $\theta_0$. We prove a Plancherel-Harish-Chandra type formula for this orbital integral, namely as an integral over the irreducible tempered auto-dual representations of $G$ that we call "symplectic" (meaning their Langlands parameter factors through $\text{Sp}_{2n}(\mathbb{C})$). This solves a problem raised by G. Chenevier and L. Clozel. Our method uses the endoscopic transfer to $\text{SO}_{2n+1}$. Along the way, we also prove that the Plancherel measure is constant on $L$-packets.