Equivariant Beilinson–Bloch conjecture for unitary Shimura varieties

Let $E/F$ be a CM extension of number fields, with complex conjugation $\mathtt{c}$, and let $n=2r$ be an even positive integer. Let $W_r=E^n$ with the specified skew-hermitian form, set $G_r=\mathrm{U}(W_r)$, and let $V$ be a totally positive definite incoherent hermitian space over $\mathbb{A}_E$ of rank $n$, with $H=\mathrm{U}(V)$. For an embedding $\iota\colon E\hookrightarrow\mathbb{C}$, let $\{\prescript{\iota}{}{X_L}\}$ be the associated unitary Shimura varieties of dimension $n-1$. Let $\pi$ be a tempered cuspidal automorphic representation of $G_r(\mathbb{A}_F)$. For an irreducible admissible representation $\tilde\pi^\infty$ of $H(\mathbb{A}_F^\infty)$ satisfying the two stated local matching and cohomological Hom conditions, let $\Pi_{j(\tilde\pi^\infty)}$ be the cuspidal factor of the automorphic base change determined by $\tilde\pi^\infty$. Equivariant Beilinson–Bloch conjecture. One has

$$\dim_\mathbb{C}\operatorname{Hom}_{H(\mathbb{A}_F^\infty)}\left(\tilde\pi^\infty,\varinjlim_L\operatorname{CH}^r(\prescript{\iota}{}{X_L})^0_\mathbb{C}\right)=\operatorname{ord}_{s=\frac12}L(s,\Pi_{j(\tilde\pi^\infty)}).$$

This is the unitary-Shimura-variety refinement of the Beilinson–Bloch conjecture, predicting Chow-group multiplicities from central L-function orders. The source does not state a resolution.