Conjecture on smooth Kähler structures for formally Kähler manifolds

Let $p\gg 0$, and let $X$ be a $W^{2,p}\times W^{1,p}$ formally Kähler manifold with almost-complex structure $J$ and nearby form $\omega_J$. Smooth Kähler structure conjecture. $X$ has a smooth structure compatible with $J$ such that there is a symplectic form smooth for this structure, and $X$ is $W^{2,p}$ diffeomorphic to a smooth Kähler manifold. This would subsume the applications of the Kähler thresholds by asserting that the limiting manifold admits a smooth Kähler structure; the conjecture is presented as a direction for future work.