Dixmier's unitarizability conjecture

Let $G$ be a countable group. A representation of $G$ on a Hilbert space is uniformly bounded if there is a constant $K$ such that $\lvert \pi(g)\rvert<K$ for all $g\in G$, and $G$ is unitarizable if every uniformly bounded representation of $G$ is unitarizable. Dixmier's conjecture. A countable group is unitarizable if and only if it is amenable. This is a fundamental problem relating bounded representations to amenability; the paper studies non-amenable wreath products and proves that they are not strongly Ulam stable, but does not resolve the conjecture in general.