Whittaker-function formula for the top mixed intersection number

Let $x_1,\dots,x_n,y_1,\dots,y_n$ be a basis of the hermitian space ${\mathbb V}$, and let $\mathbf{x}=(x_1,\dots,x_n)$ and $\mathbf{y}=(y_1,\dots,y_n)$. Define

$$B=\begin{pmatrix}(h(x_i,x_j))&(h(x_i,y_l))\\(h(y_k,x_j))&(h(y_k,y_l))\end{pmatrix}_{1\leq i,j,k,l\leq n}.$$

Let ${\mathcal I}_n^n(\mathbf{x},\mathbf{y})$ be the corresponding arithmetic intersection number, and let $W_{n,t}(B,0)$, $W'_{n,n}(B,0)$, $\beta_i^n$, and $A_n=\operatorname{diag}(1_n,\pi^{-1}1_n)$ be as in the paper. Top mixed-intersection formula. One has

$${\mathcal I}_n^n(\mathbf{x},\mathbf{y})=\frac{1}{W_{n,n}(A_n,0)}\left\{W'_{n,n}(B,0)-\sum_{0\leq i\leq n-1}\beta_i^nW_{n,i}(B,0)\right\}.$$

This is the $h=n$ instance of the broader mixed special-cycle conjecture. The source gives no resolution for this formula.