Injectivity of the spherical mean operator on the conical manifolds of spheres

Let $f$ be a continuous function on $\mathbb{R}^n$. If $f$ has zero integral over every sphere intersecting a given subset $A$ of $\mathbb{R}^n$ and $A$ lies in no affine plane of dimension $n-2$, then $f$ vanishes identically. The condition on the dimension of $A$ is sharp.