Kleppe–Ellia's stable degeneracy conjecture for curves on cubic surfaces

Let $C \subset S \subset \mathbb P^3$ be a smooth connected curve on a smooth cubic surface, and suppose that $\chi(\mathbb P^3,\mathcal I_C(3))\geq 1$. The curve $C$ is linearly normal if its embedding in $\mathbb P^3$ has the expected restriction on global sections.

Kleppe–Ellia's conjecture. If $C$ is linearly normal, then every small deformation $C'$ of $C$ in $\mathbb P^3$ is contained in a cubic surface $S'\subset\mathbb P^3$; equivalently, $C$ is stably degenerate.

This is a special case of the problem of determining when deformations of curves on surfaces remain contained in algebraically equivalent surfaces. The source explains that the unrestricted version fails for linearly non-normal curves, while the linearly normal formulation is the proposed restriction; its resolution is not established in the supplied text.