Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the hypersurface of the same degree containing $H$. In particular $X_2$ and $X_3$ are smooth.

Question: If $n = 5$ is $X_{2,3}$ necessarily singular? Is $X_{2,3}$ smooth for $n\geq 6$?