On reflection of stationary sets in $\mathcal {P}_\kappa \lambda$

Let $\kappa$ be an inaccessible cardinal, and let $E_{0} = \{x \in \mathcal {P}_{\kappa }\kappa ^{+} : \text {cf} \; \lambda _{x} = \text {cf} \; \kappa _{x}\}$ and $E_{1} = \{x \in \mathcal {P}_{\kappa }\kappa ^{+} : \kappa _{x}$ is regular and $\lambda _{x} = \kappa _{x}^{+}\}$. It is consistent that the set $E_{1}$ is stationary and that every stationary subset of $E_{0}$ reflects at almost every $a \in E_{1}$.