Uniqueness conjecture for relatively free sections of del Pezzo fibrations

Let $\pi:\mathcal{X}\to B$ be a smooth del Pezzo fibration. Fix an intersection profile $\lambda$, let $N_\lambda$ be the affine subset of $N_1(\mathcal{X})$ consisting of curve classes with the prescribed intersections with $\pi$-vertical divisors, and define

$$\operatorname{Nef}_\lambda=\operatorname{Nef}_1(\mathcal{X})\cap N_\lambda.$$

Let $\mathcal{T}$ be a translate of $\operatorname{Nef}_\lambda$ in $N_\lambda$, and write $\mathcal{T}_{\mathbb Z}$ for its integral classes. Uniqueness conjecture. There is such a translate $\mathcal{T}$ that every class in $\mathcal{T}_{\mathbb Z}$ is represented by exactly $|\operatorname{Br}(\mathcal{X})|$ different families of relatively free sections. The statement is a precise interpretation of an earlier principle asserting eventual uniqueness up to the Brauer-group factor; the source gives no resolution evidence.