Complex analyticity of Gromov–Hausdorff limits

Let $(X_j,\omega_j,p_j)$ be a sequence of $n$-dimensional Kähler–Einstein manifolds with $\operatorname{Ric}(\omega_j)=c_j\omega_j$, $|c_j|\leq 1$, and $\operatorname{Vol}(B(p_j,1))\geq\kappa>0$, converging in the Gromov–Hausdorff sense to $(X_\infty,d_\infty,p_\infty)$. Write $X_\infty=X_\infty^{\mathrm{reg}}\cup X_\infty^{\mathrm{sing}}$ for the Cheeger–Colding regular-singular decomposition, and let $\mathcal O$ be the push-forward to $X_\infty$ of the structure sheaf of $X_\infty^{\mathrm{reg}}$. Complex analyticity conjecture. $(X_\infty,\mathcal O)$ is a normal complex-analytic space, and its singular set in the complex-analytic sense coincides with $X_\infty^{\mathrm{sing}}$. This is posed as an important open question when no polarization is available; the source gives no resolution.