Conjectural leading constant in the all-height Manin conjecture

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $c(\mathcal X,\lambda)$ be the leading constant in the all-height Manin conjecture. Let $\mathsf B$ be the number of algebraic equivalence classes representing a single numerical class. Conjectural leading constant in the all-height Manin conjecture.

$$c(\mathcal X,\lambda)=\frac{\beta(\mathcal X_{\eta})}{\mathsf B}\int_{\mathcal X_{\eta}(\mathbb A_{K(B)})^{\operatorname{Br}(\mathcal X_{\eta})}_{\lambda}}\,\mathrm d\tau_{\mathcal X}.$$

The source additionally asks whether $\mathsf B=\#\operatorname{Br}(\mathcal X)$; this proposed formula is presented as conjectural and unresolved.