Coamenability and cospectral radius for orbit equivalence relations

We consider inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving orbit equivalence relations. In previous work with Ab\'{e}rt-Fra\c{c}zyk, we established the pointwise almost sure existence of the cospectral radius of a random walk on the $\mathcal{R}$-classes. In this paper, we investigate the connections of this cospectral radius to the coamenability of the inclusion $\mathcal{S}\leq \mathcal{R}$. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.