Let $P, Q$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Assume that $pr_2(Hilb_{Q,P})$ is positive dimensional where $pr_2$ is the natural projection map onto the second coordinate and $Hilb_{Q,P}$ is the flag Hilbert scheme parametrizing pairs of curves $(C_1, C_2)$ with $C_1 \in Hilb_Q, C_2 \in Hilb_P$ and $C_1 \subset C_2$. Assume further that the Hilbert scheme $Hilb_P$ is non-reduced at every point in the sense that the local ring at every point of the Hilbert scheme contains nilpotent elements. Is it then true that $pr_2(Hilb_{Q,P})$ is non-reduced as well?
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