Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $H_x\subset G$ be the stability subgroup at $x$.

**Question:** Does there exist a toral subgroup $\mathbb{T}\subset G$ such that the subset $\{x\in G/H\,|\quad\mathbb{T}\cap H_x\neq\{1\}\}$ is at most countable?

Here $\mathbb{T}\subset G$ is a closed subgroup isomorphic to the usual torus $U(1)$.

This may turn out to be a very easy question, I just can't figure out an efficient approach. Thank you.