Weak adjunction conjecture for log canonical pairs

Let $f\colon X\to Z$ be a fiber space and let $D$ be a $\mathbb{Q}$-divisor on $X$ such that $(X,D)$ is log canonical and $K_X+D$ is $\mathbb{Q}$-linearly trivial over $Z$. Let $D_Z$ denote the discriminant divisor on $Z$. Weak adjunction conjecture. There is an effective $\mathbb{Q}$-divisor $M_Z$ on $Z$ such that $K_Z+D_Z+M_Z$ is log canonical and

$$K_X+D\sim_{\mathbb{Q}}f^*(K_Z+D_Z+M_Z).$$

This is the weak form of the adjunction conjecture used in the paper's main inequality. The supplied text notes that it holds when $K_X+D$ has a Kawamata log terminal numerical complement, but gives no general resolution.