Conjecture on quasi-log schemes and short affine-line curves

Let $[X,\omega]$ be a quasi-log scheme and let $ipi\colon X\to S$ be a projective morphism between schemes such that $-\omega$ is $ipi$-ample and the induced morphism from the non-qlc locus is finite. Let $P$ be a closed point of $S$ for which there is a curve $C^\dag\subset ipi^{-1}(P)$ meeting the non-qlc locus. The conjecture. There exists a non-constant morphism

$$f\colon \mathbb A^1\longrightarrow \left(X\setminus \operatorname{Nqklt}(X,\omega)\right)\cap ipi^{-1}(P)$$

whose image closure $C$ meets the non-qlc locus and satisfies

$$0<-\omega\cdot C\leq 1.$$

This would provide a short rational curve in the fibre avoiding the non-qlc locus except at its boundary; its resolution status is not specified in the source.