Beilinson's Arakelov height-pairing conjecture for arithmetic varieties

Let $R$ be a regular arithmetic ring, let $B_0$ be a regular integral scheme flat and of finite type over $R$, and let $X_0$ be a smooth projective integral variety of dimension $d$ over the function field $K_0$ of $B_0$. Let $\widehat{\rm Pic}(B_0)$ be the group of hermitian line bundles, and let $\widehat{\rm CH}^p_{\rm hom}(X_0)_{\bf Q}$ denote the homologically trivial codimension-$p$ cycles with rational coefficients. Arakelov height-pairing conjecture. If $p+q=d+1$, there exists a pairing

$$h:\operatorname{CH}^p_{\rm hom}(X_0)_{\bf Q}\otimes \operatorname{CH}^q_{\rm hom}(X_0)_{\bf Q}\to \widehat{\operatorname{Pic}}(B_0)_{\bf Q}$$

with the following properties: in the number-field case over $\operatorname{Spec}R$, its arithmetic degree is Beilinson's conjectural height pairing; when $B_{k_0}$ is geometrically integral, its image under the forgetful base-change map agrees with the pairing of the motivic-origin conjecture; and, if $B_0$ is projective, the resulting real-valued pairing is expected to be non-degenerate after composition with the arithmetic degree map associated with an ample hermitian line bundle. The source presents this as a conjectural Arakelovian counterpart; existence and the stated compatibility and non-degeneracy properties are not established in general.