Classification of reductive homogeneous spaces satisfying strict inequality for Benoist-Kobayashi's $\rho$ functions

Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G\mathrm{/}H)$ is a tempered representation of $G$. They introduced the functions $\rho$ on Lie algebras and gave a necessary and sufficient condition for the temperedness of $L^2(G\mathrm{/}H)$ in terms of an inequality on $\rho$. In a joint work with Y. Oshima, we considered when $L^2(G\mathrm{/}H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and gave a sufficient condition for this in terms of a strict inequality of $\rho$. In this paper, we will classify the pairs $(\mathfrak{g},\mathfrak{h})$ with $\mathfrak{g}$ complex reductive and $\mathfrak{h}$ complex semisimple which satisfy that strict inequality of $\rho$.