The modified Kodaira conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold. A compact Kähler space is algebraically approximable if it occurs as the central fibre of a proper flat family having projective fibres along a sequence converging to the central parameter. A bimeromorphic model is a space $X'$ bimeromorphic to $X$.

Modified Kodaira conjecture. There exists a bimeromorphic model

$$X\dashrightarrow X'$$

where $X'$ is a normal ${\mathbb{Q}}$-factorial compact Kähler space with terminal singularities and is algebraically approximable.

This modifies Kodaira’s question by allowing singular minimal-model-type bimeromorphic models. Smooth compact Kähler manifolds can fail to have algebraically approximable smooth bimeromorphic models in higher dimensions, while the source reports a partial positive result in dimension three when $\kappa(X)=0$.