Kleppe–Ellia conjecture on cubic-surface space-curve families

Let $H(d,g)^{sc}$ denote the Hilbert scheme of smooth connected curves in $\mathbb P^3$ of degree $d$ and genus $g$, and let $W\subset H(d,g)^{sc}$ be a $3$-maximal family whose general member $C$ lies on a smooth cubic surface. Let $\mathcal I_C$ be the ideal sheaf of $C$ in $\mathbb P^3$. Suppose that

$$d>9,\qquad g\geq 3d-18,$$

that $C$ is linearly normal, and that

$$H^1(\mathbb P^3,\mathcal I_C(3))\neq 0.$$

Kleppe–Ellia conjecture. The family $W$ is an irreducible component of $(H(d,g)^{sc})_{\operatorname{red}}$, and $H(d,g)^{sc}$ is generically non-reduced along $W$.

This conjecture concerns the component structure and generic non-reducedness of the Hilbert scheme along families of space curves on smooth cubic surfaces. It is presented as a version of a conjecture originated by Kleppe and modified by Ellia; the supplied text does not state whether it has been resolved.