  We show that the manifold $(\mathbb{S}^2 \times \mathbb{S}^2)
\operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$ does not admit a
non-constant non-injective uniformly quasiregular self-map. This answers a
question of Martin, Mayer, and Peltonen, and provides the first example of a
quasiregularly elliptic manifold which is not uniformly quasiregularly
elliptic.
  To obtain the result, we introduce conformally formal manifolds, which are
closed smooth $n$-manifolds $M$ admitting a measurable conformal structure
$[g]$ for which the $(n/k)$-harmonic $k$-forms of the structure $[g]$ form an
algebra. This is a conformal counterpart to the existing study of geometrically
formal manifolds. We show that, similarly as in the geometrically formal
theory, the real cohomology ring $H^*(M; \mathbb{R})$ of a conformally formal
$n$-manifold $M$ admits an embedding of algebras $\Phi \colon H^*(M;
\mathbb{R}) \hookrightarrow \wedge^* \mathbb{R}^n$. We also show that uniformly
quasiregularly elliptic manifolds $M$ are conformally formal in a stronger
sense, in which the wedge product is replaced with a conformally scaled
Clifford product. For this stronger version of conformal formality, the image
of $\Phi$ is closed under the Euclidean Clifford product of $\wedge^*
\mathbb{R}^n$, which in turn is impossible for $M = (\mathbb{S}^2 \times
\mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$.
