Regular semisimple Hessenberg varieties with cohomology rings generated in degree two

A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the flag variety determined by a square matrix $S$ with distinct eigenvalues and a Hessenberg function $h$. The cohomology ring $H^*(\mathrm{Hess}(S,h))$ is independent of the choice of $S$ and is not explicitly described except for a few cases. In this paper, we characterize the Hessenberg function $h$ such that $H^*(\mathrm{Hess}(S,h))$ is generated in degree two as a ring. It turns out that such $h$ is what is called a (double) lollipop.