Let $\kappa$ be an infinite cardinal. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, denote by $b({\cal U})$ the minimal size that a filter base for ${\cal U}$ can have.

If ${\cal U, V}$ are non-principal ultrafilters on $\kappa$, do we necessarily have $b({\cal U}) = b({\cal V})$?