  Using a probabilistic argument we show that the second bounded cohomology of
an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or
relatively hyperbolic group, non-exceptional mapping class group, ${\rm
Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse
limit of the second bounded cohomologies of its virtually free subgroups, and
in fact even into the inverse limit of the second bounded cohomologies of its
hyperbolically embedded virtually free subgroups. This result is new and
non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The
corresponding statement fails in general for the third bounded cohomology, even
for surface groups.
