Antipodal paths in covers of spheres

In this note we show that if the sphere $\mathbb{S}^n$ is covered by $k$ open sets with $n \geq 2k-2$, then one of these sets contains a path with antipodal endpoints.

This is best possible in the sense that the statement fails for $n < 2k-2$.

The result can be seen as a spherical analogue of a well-known conjecture of Norine on edge-colourings of the discrete hypercube.