  A probability-measure-preserving transformation has the Weak Pinsker Property
(WPP) if for every $\epsilon>0$ it is measurably conjugate to the direct
product of a transformation with entropy $<\epsilon$ and a Bernoulli shift. In
a recent breakthrough, Tim Austin proved that every ergodic transformation
satisfies this property. Moreover, the natural analog for amenable group
actions is also true. By contrast, this paper provides a counterexample in
which the group $\Gamma$ is a non-abelian free group and the notion of entropy
is sofic entropy. The counterexample is a limit of hardcore models on random
regular graphs. In order to prove that it does not have the WPP, this paper
introduces new measure conjugacy invariants based on the growth of homology of
the model spaces of the action. The main result is obtained by showing that any
action with the WPP has subexponential homology growth in dimension $0$, while
the counterexample has exponential homology growth in dimension $0$.
