  Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for
$n\ge 2$, which is not a rational homology sphere. We show that, for a
non-constant and non-injective uniformly quasiregular self-map $f\colon M\to
M$, the topological entropy $h(f)$ is $\log \mathrm{deg}( f )$. This proves
Shub's entropy conjecture in this case.
