B-semiampleness for compact log-smooth Kähler fibrations

We prove the compact log-smooth Kähler case of b-semiampleness. Let $f:Y\to X$ be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in $[0,1]$, with $K_Y+\Delta {\sim }_{\mathbb{Q}}{f}^{\ast }L$ for $L\in \mathrm{Pic}(X{)}_{\mathbb{Q}}$. There is a smooth compact Kähler modification $S\to X$ for which the threshold-moduli line satisfies $M_{S_1}={\nu }^{\ast }{M}_S$ in $\mathrm{Pic}(S_1{)}_{\mathbb{Q}}$ for every smooth compact Kähler modification $\nu :S_1\to S$, and some positive multiple of M${}_S$ is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.