If $(X,\tau)$ is a $T_2$-space such that [all non-empty open sets are homeomorphic](https://mathoverflow.net/questions/300253/t-2-spaces-where-all-non-empty-open-sets-are-homeomorphic) (with the subspace topology) to $X$, is $(X,\tau)$ necessarily [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_space)?