Arithmetic level raising, self-dual case

In the indefinite case, let $\mathcal{S}_0^\circ$ be the relevant Shimura variety, let $\widetilde{\mathcal{S}}_0^\circ$ be its special fiber, let $\mathrm{S}'_0$ parameterize the irreducible components of its basic locus, and let $\mathfrak{n}_0$ be the localization ideal. Let $n$, $r$, $\lambda$, $\Pi_{0,\mathfrak{p}}$, and $\mathrm{V}^\Pi_{0,\lambda}$ be as in the source. Arithmetic level raising, self-dual case. If (a) $\mathrm{H}^i(\mathcal{S}_0^\circ\otimes_F\overline{\mathbb{Q}},\mathbb{Z}_\lambda)_{\mathfrak{n}_0}=0$ for $i\neq n-1$ and $\mathrm{H}^{n-1}(\mathcal{S}_0^\circ\otimes_F\overline{\mathbb{Q}}_p,\mathbb{Z}_\lambda)_{\mathfrak{n}_0}$ is finite free over $\mathbb{Z}_\lambda$, (b) $\mathrm{V}^\Pi_{0,\lambda}$ is residually absolutely irreducible, and (c) the Satake parameter of $\Pi_{0,\mathfrak{p}}$ modulo $\lambda$ contains $p$ at most once, then the localized absolute cycle class map

$$\mathbb{Z}_\lambda[\mathrm{S}'_0]_{\mathfrak{n}_0}\longrightarrow \mathrm{H}^{2r}(\widetilde{\mathcal{S}}_0^\circ,\mathbb{Z}_\lambda(r))_{\mathfrak{n}_0}$$

is surjective. This generalizes Ribet’s level-raising theorem for modular curves, or the corresponding unitary Shimura curves, at good places; the assertion itself is presented as a conjecture.