The lcK existence conjecture for closed smooth four-manifolds

Let $X$ be a closed smooth four-manifold. The lcK existence conjecture. If $X$ admits a complex structure, then $X$ also admits a locally conformally Kähler (lcK) structure, not necessarily compatible with the given complex structure. This conjecture predicts that in dimension four, unlike in higher dimensions, the existence of a complex structure should ensure the existence of an lcK structure; the source presents it as very likely because no examples of the higher-dimensional phenomenon are known in dimension four.