The commutativity conjecture for the identity component of a pseudofinite group's Bohr compactification

Let $G$ be a pseudofinite group, meaning an infinite group elementarily equivalent to an ultraproduct of finite groups, and let $bG$ denote its Bohr compactification. Write $(bG)^{0}$ for the connected component of the identity in $bG$.

Commutativity conjecture. If $G$ is a pseudofinite group, then $(bG)^{0}$ is commutative.

This conjecture is equivalent to a negative answer to the paper's Question 1.4, modulo passing to connected components and allowing a finite product of compact simple Lie groups in place of a single one. Its status is not resolved in the supplied source.