Conjectural low-regularity Newlander–Nirenberg theorem

Let $X$ be a $W^{2,p}\times W^{1,p}$ formally Kähler manifold with complex structure $J$ and metric $g$. Low-regularity Newlander–Nirenberg conjecture. The Newlander–Nirenberg theorem on integrability of the complex structure holds, yielding holomorphic coordinate charts and a new smooth structure in which $J$ is smooth while $g$ remains $W^{2,p}$. This is identified as a basic ingredient needed to establish the conjectured smooth Kähler structure; its resolution is not given in the source.