Let $P_1, P_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme $\mathrm{Hilb}_{P_1,Q}$ to $\mathrm{Hilb}_{P_2,Q}$ under the map, $(C,X) \mapsto (2C,X)$. Then is the image under the composition of the maps, $i$ with the natural projection map $\mathrm{pr}_2$,

$\mathrm{Hilb}_{P_1,Q} \to\mathrm{Hilb}_{P_2,Q}\to \mathrm{Hilb}_{Q}$

an irreducible component of the image of $\mathrm{pr}_2$? If so can this result be generalized to the case when we can replace plane conic and quartic by curves $C_1, C_2$ such that $rC_1$ has the same Hilbert polynomial as $C_2$?

Note:Hilbert flag scheme $\mathrm{Hilb}_{P_i,Q}$ parametrize pairs of the form $C \subset X$ where $P_i$ is the Hilbert polynomial of $C$ and $X$ is a degree $d$ surface in $\mathbb{P}^3$.