Deformation conjecture for the specialness invariant

Let $X\in\mathcal C$ be smooth, and let $\operatorname{ess}(X)$ denote the specialness invariant introduced in the paper. Deformation conjecture. The invariant $\operatorname{ess}(X)$ is invariant under deformation. In particular, the class of special manifolds is stable under deformation. The source gives no resolution.