On the canonical bundle formula in positive characteristic

Let $f:X\to Z$ be a fibration from a normal projective variety $X$ of dimension $n$ onto a normal curve $Z$ over a perfect field of characteristic $p>2$. Let $(X,B)$ be a dlt pair such that the induced pair on a general fibre is log canonical. Assuming the LMMP and the existence of log resolutions in dimension $\le n$, we prove that, when $K_X+B$ is $f$-nef, the moduli part is nef up to a birational map $Y⇢X$. As a corollary, we prove positivity of the moduli part in the $K$-trivial case, i.e. when $K_X+B{\sim }_{\text{Q}}{f}^{\ast }L$ for some $\text{Q}$-Cartier $\text{Q}$-divisor $L$ on $Z$. In particular, consider a dlt pair $(X,B)$ of dimension $3$ over an algebraically closed field of characteristic $p>5$ such that the induced pair on a general fibre is log canonical, then the canonical bundle formula holds unconditionally.