  We show that for any infinite countable group $G$ and for any free Borel
action $G \curvearrowright X$ there exists an equivariant class-bijective Borel
map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G
\curvearrowright 2^G$. This implies that any Borel structurability which holds
for the equivalence relation generated by $G \curvearrowright
\mathrm{Free}(2^G)$ must hold a fortiori for all equivalence relations coming
from free Borel actions of $G$. A related consequence is that the Borel
chromatic number of $\mathrm{Free}(2^G)$ is the maximum among Borel chromatic
numbers of free actions of $G$. This answers a question of Marks. Our
construction is flexible and, using an appropriate notion of genericity, we are
able to show that in fact the generic $G$-equivariant map to $2^G$ lands in the
free part. As a corollary we obtain that for every $\epsilon > 0$, every free
pmp action of $G$ has a free factor which admits a $2$-piece generating
partition with Shannon entropy less than $\epsilon$. This generalizes a result
of Danilenko and Park.
