Let $P$ (resp. $Q$) be the Hilbert polynomial of a curve in $\mathbb{P}^3$ (resp. a degree $d$ surface in $\mathbb{P}^3$). Denote by $Hilb_{P,Q}$ the flag Hilbert scheme corresponding to the pair of Hilbert polynomials $P,Q$. Assume that the image under the second projection map $\mathrm{pr}_2(Hilb_{P,Q})$ is at least $2$-dimensional. Under what condition can we conclude that for a general element $(C,X)$ in $Hilb_{P,Q}$, there exists an open set $U$ in the linear system $|C|$ such that the intersection of the spaces $\cap_{D \in U} I_d(D)$ is at least $2$ dimensional?

Note: For a curve $D$, $I_d(D)$ denotes the graded $d$ piece of the ideal of $D$.