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 $b({\cal U}) = b({\cal V})$?

Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be uniform if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, denote by $b({\cal U})$ the minimal cardinality that a filter base for ${\cal U}$ can have.

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

Thanks to Joseph van Name for making me aware of uniform ultrafilters and the fact that this question is only (potentially) interesting when restricted to these.