Let $H$,$K$ be closed connected subgroups of a compact Lie group $G$. Let $L:=\langle H,K \rangle$ be the subgroup they generate, ie, the smallest subgroup of $G$ containing them both. Must $L$ be closed?

Notes:

1. False if $G$ is not compact.

2. False if $H$, $K$ not connected: consider two $\mathbb{Z}/2\mathbb{Z}$ subgroups in $O(2)$ generated by reflections in irrationally related axes.

Is there a way to jack up (2) to a connected counterexample, for instance?