Relative Clemens conjecture for rational curves on cubic threefolds

Let $X\subset\mathbb{P}^4$ be a cubic threefold, let $Y\subset X$ be a hyperplane, and let $C\subset X$ be a smooth rational curve of degree $d$, with $X$, $Y$, and $C$ all general. Let $Z=C\cap Y$ and let $N_{C/X}$ denote the normal bundle of $C$ in $X$. Relative Clemens conjecture. The threefold $X$ admits only a finite number of general rational curves whose intersection with $Y$ is $Z$. Moreover, every such curve has normal bundle

$$N_{C/X}(-Y)\cong\mathscr{O}_C(-1)\oplus\mathscr{O}_C(-1).$$

This is the relative analogue of the Clemens conjecture, prescribing finiteness while fixing the intersection with the hyperplane. The supplied passage gives no resolution status for the assertion.