Let $C_1 \cup C_2$ be a curve in $\mathbb{P}^3$ and $X$ be a smooth degree $d$ surface in $\mathbb{P}^3$ containing them and $d \ge 6$. Further, assume that the minimum degree polynomial in $I(C_1 \cup C_2)$ is of degree less than $d/2$. Is it true that there exists a smooth degree $d$ surface in $\mathbb{P}^3$ containing $C_1$ and a line? (The underlying field is always $\mathbb{C}$)
