The Burger-Ozawa-Thom question: does strong Ulam stability characterize amenability?

Theorem 1.1: a countable discrete group is strongly Ulam stable if and only if it is amenable. For every nonamenable countable $G$ and $\delta>0$ there is a separable $H$ and a normalized $\mu:G\to U(H)$ with defect below $\delta$ that stays at uniform distance more than $1/10$ from every unitary representation on $H$. Not shown: anything about finite-dimensional (ordinary) Ulam stability, or uncountable groups.

Theorem 1.1: a countable discrete group is strongly Ulam stable if and only if it is amenable. For every nonamenable countable $G$ and $\delta>0$ there is a separable $H$ and a normalized $\mu:G\to U(H)$ with defect below $\delta$ that stays at uniform distance more than $1/10$ from every unitary representation on $H$. Not shown: anything about finite-dimensional (ordinary) Ulam stability, or uncountable groups.