The log canonical adjunction, effective log abundance, and effective adjunction conjectures

Let $f\colon X\to Z$ be the fibre space and let $D$, $\Theta$, $D_{\operatorname{div}}$, and $D_{\operatorname{mod}}$ be as in the paper's adjunction construction. Assume that $D$ and $\Theta$ are $\mathbb{Q}$-divisors, $K_X+\Theta$ is $\mathbb{Q}$-linearly trivial over $Z$, and $(F,(1-t)D|_F+t\Theta|_F)$ is klt for every $0<t\leq 1$, where $F$ is the generic fibre. In particular, $D_{\operatorname{div}}$ and $D_{\operatorname{mod}}$ are $\mathbb{Q}$-divisors. Adjunction conjectures. The following assertions are expected: $D_{\operatorname{mod}}$ is b-semiample; if $X_\eta$ is the generic fibre of $f$, then $I_0(K_{X_\eta}+D_\eta)\sim 0$ for an integer $I_0$ depending only on $\dim X_\eta$ and the multiplicities of $D^{\mathrm h}$; and $D_{\operatorname{mod}}$ is effectively b-semiample, meaning that there is a positive integer $I$, depending only on the dimension of $X$ and the horizontal multiplicities of $D$, such that $I D_{\operatorname{mod}}$ is very b-semiample, namely $I D_{\operatorname{mod}}=\overline{M}$ for a base-point-free divisor $M$ on some model $Z'/Z$. These assertions concern the expected positivity and effective control of the moduli part in canonical bundle adjunction; the cited theorem preceding them establishes only b-nefness under different klt hypotheses, so the conjectural semiampleness and effective bounds remain open in the supplied text.