Are the dot products of all vector spherical harmonics complete?

Question

Does the set of all dot products ${\bf Y}^j_{jm} (\theta, \phi) \cdot {\bf Y}^{j'}_{j'm'}(\theta,\phi)$, where ${\bf Y}^j_{jm}$ are vector spherical harmonics ($j,j' = 0,1,2, ...$ and $m,m' = -j, ..., 0, ... j$), span the same space as the scalar spherical harmonics? Do you have a proof either way?