Itzkowitz's problem for group of finite exponent

Itzkowitz's problem asks whether every topological group $G$ has equal left and right uniform structures provided that bounded left uniformly continuous real-valued function on $G$ are right uniformly continuous. This paper provides a positive answer to this problem if $G$ is of bounded exponent or, more generally, if there exist an integer $p\geq 2$ and a nonempty open set $U\subset G$ such that the power map $U\ni g\to g^p\in G$ is left (or right) uniformly continuous. This also resolves the problem for periodic groups which are Baire spaces.