The Abel–Jacobi injectivity conjecture for special cycles

Let $\rho^{?}\in\{\rho,\rho^*(1)\}$, let $K^p\subset \mathrm{H}(\mathbf{A}^{p\infty})$ be open compact, and let $K_p^\circ$ be the chosen hyperspecial subgroup. Let $\mathfrak{m}_{\rho^{?}}$ be the corresponding maximal ideal, let $\mathrm{Ch}^r(X_{K^pK_p^\circ})_{\overline{\mathbf{Q}}{}_p}^{\langle p\rangle}$ be the relevant localized cycle space, and let $M_{\rho^{?},K^pK_p^\circ}$ be the associated Galois representation. Abel–Jacobi injectivity conjecture. The Abel–Jacobi map

$$\operatorname{AJ}_{p,K^pK_p^\circ}\colon \left(\mathrm{Ch}^r(X_{K^pK_p^\circ})_{\overline{\mathbf{Q}}{}_p}^{\langle p\rangle}\right)_{\mathfrak{m}_{\rho^{?}}}\longrightarrow H^1_f(E,M_{\rho^{?},K^pK_p^\circ})$$

is injective for both choices $\rho^{?}\in\{\rho,\rho^*(1)\}$ and every open compact $K^p$. This injectivity is assumed in order to descend the Beilinson–Bloch height pairing from cycles to Selmer groups. The source gives no evidence of a general resolution.