Mixed-characteristic density conjecture for semiampleness

Let $R$ be a finitely generated integral domain over $\mathbb Z$, let $S=\operatorname{Spec}R$, and let $\pi:\mathcal X\to S$ be a smooth proper morphism. Let $\mathcal L$ be an invertible sheaf on $\mathcal X$, and assume that its restriction to the generic geometric fibre of $\pi$ is nef. Mixed-characteristic semiampleness conjecture. The set $T$ of closed points $s\in S$ for which $\mathcal L_s$ is semiample is dense in $S$. The conjecture proposes that the variation of nefness seen in positive equal characteristic does not occur in this mixed-characteristic setting.