An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap_{\alpha \in y} X_\alpha = \emptyset$. Countable incompleteness is equivalent to $(\omega,\omega)$-regularity, and for every $\kappa$, there is always an $(\omega,\kappa)$-regular ultrafilter. The failure of $(\omega,\omega_1)$-regularity for an ultrafilter on $\omega_1$ is consistent relative to large cardinals, and carries some large cardinal strength.

In the 1970s, Kanamori proved that if $\lambda$ is singular, then every ultrafilter on $\lambda^+$ is $(\lambda,\lambda^+)$-regular. **Question:** Is it known whether Kanamori's result can be improved to, "If $\lambda$ is singular, then every ultrafilter on $\lambda^+$ is $(\kappa,\lambda^+)$-regular for some $\kappa < \lambda$"? What about in the case $\lambda=\aleph_\omega$?

More on regular and decomposable ultrafilters in ZFC. $\endgroup$