Singularities on Vertical $\epsilon $-Log Canonical Fano Fibrations

Given a Fano type log Calabi–Yau fibration $(X,B)\to Z$ with $(X,B)$ being $\epsilon $-lc, the first author in 2023 proved that the generalised pair $(Z,B_Z+M_Z)$ given by the canonical bundle formula is generalised $\delta $-lc, where $\delta >0$ depends only on $\epsilon $ and $\dim X-\dim Z$, which confirmed a conjecture of Shokurov. In this paper, we prove the above result under a weaker assumption. Instead of requiring $(X,B)$ to be $\epsilon $-lc, we assume that $(X,B)$ is $\epsilon $-lc vertically over $Z$, that is, the log discrepancy of $E$ with respect to $(X,B)$ is ${\geq }\,\epsilon $ for any prime divisor $E$ over $X$ whose centre on $X$ is vertical over $Z$.