Pullback description conjecture for movable classes annihilated by a semiample-type class

Let $\varphi:X\rightarrow Y$ be a surjective holomorphic map between compact Kähler manifolds of dimensions $n$ and $k$, respectively, and let $A$ be a Kähler class on $Y$. Set $L=\varphi^*A$. For a movable class $\alpha$ on $X$ satisfying $L^k\cdot\alpha=0$, pullback description conjecture. There should exist compact Kähler manifolds $X'$ and $Y'$, bimeromorphic holomorphic maps $\pi:X'\rightarrow X$ and $\mu:Y'\rightarrow Y$, a holomorphic map $\varphi':X'\rightarrow Y'$, and a pseudoeffective class $\beta$ on $Y'$ such that

\begin{tikzcd} X' \arrow{r}{\pi} \arrow{d}{\varphi'} & X \arrow{d}{\varphi} \\ Y' \arrow{r}{\mu}& Y \end{tikzcd}

and

$$\alpha=\pi_*\langle\varphi'^*\beta\rangle.$$

This asks whether the geometric description known in the semiample projective setting extends to the transcendental Kähler setting; the source presents it as a natural question and leaves it open.