In Alex Lee's undergraduate thesis (2000), it was said that the Hilbert scheme $H_{2m+2}(\mathbb{P}^3)$ has two components $\mathcal{H}',\mathcal{H}''$, where a general point of $\mathcal{H}'$ corresponds to a pair of skew lines and a general point of $\mathcal{H}''$ corresponds to a conic union a point. At the time of the thesis, the smoothness of neither of the two components were known.

Since then, it was shown in the paper *Hilbert scheme of a pair of codimension two subspaces* (2011) that $\mathcal{H}'$ is smooth. Is it known by now whether $\mathcal{H}''$ is smooth?