  We show that for a countable exact group, having positive first
$\ell^2$-Betti number implies proper proximality in this sense of
\cite{BoIoPe21}. This is achieved by showing a cocycle superrigidty result for
Bernoulli shifts of non-properly proximal groups. We also obtain that Bernoulli
shifts of countable, nonamenable, i.c.c., exact, non-properly proximal groups
are OE-superrigid.
