Dominant-families conjecture for sections of del Pezzo fibrations

Let $\pi:\mathcal{X}\to\mathbb{P}^{1}$ be a smooth del Pezzo fibration. Fix an intersection profile $\lambda$, meaning the prescribed intersection numbers of a section with the $\pi$-vertical divisors, and let $N_{\lambda}\subset N_{1}(\mathcal{X})$ be the affine linear subspace of curve classes with profile $\lambda$. Define

$$\mathrm{Nef}_{\lambda}=\mathrm{Nef}_{1}(\mathcal{X})\cap N_{\lambda}.$$

Let $\mathcal{T}_{\mathbb{Z}}$ denote the integral numerical classes in a translate $\mathcal{T}$ of this set.

Dominant-families conjecture. There is some translate $\mathcal{T}$ of $\mathrm{Nef}_{\lambda}$ in $N_{\lambda}$ such that every numerical class in $\mathcal{T}_{\mathbb{Z}}$ is represented by exactly $|\mathrm{Br}(\mathcal{X})|$ different dominant families of sections.

The conjecture refines the expected relationship between algebraic and numerical equivalence of curves and the Brauer group of the total space. It predicts uniform behavior only sufficiently far inside a translated nef region; no resolution is supplied in the source.