Conformally formal manifolds and the uniformly quasiregular non-ellipticity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml…

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