Conics in quintic del Pezzo varieties

The smooth quintic del Pezzo variety $Y$ is well-known to be obtained as a linear sections of the Grassmannian variety $\mathrm{G}\mathrm{r}(2,5)$ under the Plücker embedding into ${\mathbb{P}}^9$. Through a local computation, we show the Hilbert scheme of conics in $Y$ for $\text{dim}Y\ge 3$ can be obtained from a certain Grassmannian bundle by a single blowing up/down transformation.