The local intertwining relation conjecture for Mp(2n)

Let $\phi_M$ and $u$ be as in the source: $\phi_M$ is a parameter for a Levi subgroup and $u\in N_\phi:=N_{C_\phi}(A_{\tilde{M}^\vee})$ is semisimple, mapping to $w\in W_\phi$ under the natural homomorphism. Let $f^!(\phi,u)$ and $f(\phi,u)$ be the two expressions defined from these data and an anti-genuine test function $f$ on $\tilde{G}$.

The local intertwining relation conjecture. Given $\phi_M$ and $u$ as above, one expects

$$f^!(\phi,u)=f(\phi,u)$$

for all $f$. This is the proposed compatibility between the stable character expression $f^!(\phi,u)$ and the explicitly defined intertwining-operator expression $f(\phi,u)$; the source states it in the setting of the preceding constructions and leaves it as an expectation.