Regularity limit problem for singular cardinals

Let $\lambda$ be a singular cardinal, let $\mu<\lambda$, and let $D$ be an ultrafilter that is

$$(\operatorname{cf}\lambda)^+$$

-complete and $(\mu,\kappa)$-regular for every $\kappa<\lambda$. Regularity limit problem. Must $D$ be $(\mu,\lambda^+)$-regular? This asks whether the first conclusion of the relevant limit theorem remains valid when decomposability is replaced by regularity. The source presents this as a subtler problem and gives no resolution.