Saito–Kurokawa multiplicity formula for even orthogonal CAP representations

Let $F$ be a totally real number field, let $n$ be even, and let $\pi\simeq\otimes'_v\pi_v$ be an irreducible cuspidal automorphic representation of $\mathrm{PGL}_2(\mathbb A)$ with $\pi_v\simeq D_{2\kappa_v}$ at every archimedean place. For a sign character $\epsilon$ satisfying $\epsilon_v=(-1)^{n/2}$ at archimedean places, let $\mathrm{SK}_n^\epsilon(\pi)=\otimes'_v\mathrm{SK}_n^{\epsilon_v}(\pi_v)$ be the associated admissible representation of $\mathrm{PGSp}_n(\mathbb A)$. The even orthogonal CAP multiplicity conjecture. Its cuspidal multiplicity is

$$\mathrm m_\mathrm{cusp}(\mathrm{SK}_n^\epsilon(\pi))=\frac{1}{2}\left(1+\varepsilon\left(\frac{1}{2},\pi\right)\prod_v\epsilon_v\right).$$

The formula predicts the cuspidal occurrence of these CAP representations in terms of the central root number and local signs; the supplied text does not state a resolution.