Contact-form characterization of smooth isolated singularities

Let $X \subset \mathbb{C}^N$ be a variety of dimension $n$ with a normal isolated singularity at $0$. For sufficiently small $\epsilon>0$, let $L_{X,\epsilon}$ be its link, and let $\theta_{X,\epsilon}$ be the induced contact form; let $\theta_{\mathrm{st}}$ be the standard contact form on the sphere $S^{2n-1}$. Contact-form conjecture. If $L_{X,\epsilon}$ is diffeomorphic to $S^{2n-1}$ and the contact forms $\theta_{X,\epsilon}$ specialize to $\theta_{\mathrm{st}}$ as $\epsilon\to0$, then $X$ is smooth at $0$. Smooth varieties satisfy this specialization property, but the converse is proposed and remains open for normal isolated singularities.