Das–Hacon's transcendental base point free conjecture

Let $(X,B+\boldsymbol{\beta})$ be a generalized klt Kähler pair, where $X$ is a normal compact Kähler variety, and let $\alpha\in H_{\mathrm{BC}}^{1,1}(X,\mathbb{R})$ be a nef class on $X$. Das–Hacon's transcendental base point free conjecture. If

$$\alpha-(K_X+B+\boldsymbol{\beta}_X)$$

is nef and big, then there exists a proper surjective morphism with connected fibers $f:X\to Y$ to a normal compact Kähler variety $Y$ with rational singularities, and a Kähler class $\omega_Y\in H_{\mathrm{BC}}^{1,1}(Y,\mathbb{R})$, such that

$$\alpha=f^*\omega_Y.$$

The source attributes this conjecture to Das–Hacon and notes that it is used as a transcendental analogue of the base point free theorem.