  We prove that a minimizer of the Yamabe functional does not exist for a
sphere $\mathbb{S}^n$ of dimension $n \geq 3$, endowed with a standard
edge-cone spherical metric of cone angle greater than or equal to $4\pi$, along
a great circle of codimension two. When the cone angle along the singularity is
smaller than $2\pi$, the corresponding metric is known to be a Yamabe metric,
and we show that all Yamabe metrics in its conformal class are obtained from it
by constant multiples and conformal diffeomorphisms preserving the singular
set.
