The affine-action dichotomy for amenable groups

Let $G$ be an amenable group, and let $\lambda_G$ denote its left regular representation. An affine action of $G$ with linear part $\lambda_G$ has an affine action map whose linear part is $\lambda_G$. Affine-action dichotomy conjecture. Every affine action with linear part $\lambda_G$ is either bounded or proper. This is presented as a conjectural statement about affine actions on $L^2(G)$; the source does not specify whether it has been resolved.