Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ is a plane curve (in the sense the radical ideal of the curve contains a linear polynomial)?

There is another way of formulating this problem. Let $H$ be the Hilbert flag scheme of pairs $(C_1, C_2)$ with $C_1 \subset C_2$, $C_2$ is a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ and $C_1$ is of degree $e$. The question is does there exists an irreducible component $H'$ of $H$ such that a generic element of $pr_1(H')$ is a plane curve in the sense described above (Here $pr_1$ is the natural projection map onto the first coordinate)?

In the above question we can assume $C_1$ is a local complete intersection curve in $\mathbb{P}^3$.