The projective deformation-limit conjecture for Moishezon manifolds

Let $\pi:\mathcal{X}\rightarrow\Delta$ be a holomorphic family of compact complex manifolds of dimension $n$ over an open disk $\Delta\subset\mathbb{C}$, with fibers $X_t:=\pi^{-1}(t)$ and $\Delta^*:=\Delta\setminus\{0\}$. A compact connected complex manifold is Moishezon if it possesses $\dim_{\mathbb{C}}X$ algebraically independent meromorphic functions; equivalently, it admits a holomorphic modification from a projective algebraic manifold. The projective deformation-limit conjecture. If the fiber $X_t$ is projective for each $t\in\Delta^*$, then the reference fiber $X_0:=\pi^{-1}(0)$ is Moishezon. This is a longstanding deformation-limit problem; the paper studies stronger hypotheses and proves related results, while the conjecture itself is presented as open.