The refined character formula conjecture for Mp(2n)

Let $\tilde{G}=\operatorname{Mp}(2n)$, let $\Phi_{\mathrm{bdd}}(\tilde{G})$ be the set of bounded parameters, let $\Pi_\phi$ be the packet attached to $\phi$, and let $C_{\phi,\mathrm{ss}}$ be the semisimple centralizer. For $s\in C_{\phi,\mathrm{ss}}$, write $\langle s,\pi\rangle_\Theta$ for the theta-lifting character pairing and set

$$\tilde{\epsilon}_\phi(s)=\epsilon\left(\frac{1}{2},V_\phi^{s=-1},\psi\right).$$

The refined character formula conjecture. One expects

$$\langle s,\pi\rangle=\langle s,\pi\rangle_\Theta\tilde{\epsilon}_\phi(s),\qquad s\in C_{\phi,\mathrm{ss}},$$

for all $\phi\in\Phi_{\mathrm{bdd}}(\tilde{G})$ and $\pi\in\Pi_\phi$. This refines the preceding local Langlands conjecture by specifying the relation between the desired character pairing and the theta-lifting pairing; the source presents it as an expectation supported by the $n=1$ case and the global multiplicity formula.