  We show that if a compact, connected, and oriented $n$-manifold $M$ without
boundary admits a non-constant non-injective uniformly quasiregular self-map,
then the dimension of the real singular cohomology ring $H^*(M; \mathbb{R})$ of
$M$ is bounded from above by $2^n$. This is a positive answer to a dynamical
counterpart of the Bonk-Heinonen conjecture on the cohomology bound for
quasiregularly elliptic manifolds. The proof is based on an intermediary result
that, if $M$ is not a rational homology sphere, then each such uniformly
quasiregular self-map on $M$ has a Julia set of positive Lebesgue measure.
