The bimeromorphic algebraic approximation conjecture for non-uniruled compact Kähler manifolds

Let $X$ be a compact Kähler manifold that is not uniruled. A bimeromorphic map is a meromorphic map with a meromorphic inverse. An algebraic approximation is the deformation notion referred to in the source's Definition 2.1. The target $X'$ is required to be a normal compact Kähler space with terminal singularities.

Bimeromorphic algebraic approximation conjecture. There exists a bimeromorphic map

$$X \dashrightarrow X'$$

to a normal compact Kähler space $X'$ with terminal singularities that admits an algebraic approximation.

This is a flexible bimeromorphic version of Kodaira's problem, proposed for non-uniruled compact Kähler manifolds. The paper proves the assertion in the cases needed for its threefold results and establishes broader results for certain elliptic fibrations, but the full general-dimensional statement remains open.