(Everything below is assuming $V=L$.)

Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ and let $$E_\kappa^+=\{\mu<\kappa: \mbox{$\exists M\prec (L_{\kappa^+}, L_\kappa)$ with $M\cong (L_\alpha, L_\mu)$ for some $\alpha<\kappa$}\}.$$ (In each definition $\mu$ ranges over ordinals.) Here "$(L_\alpha, L_\beta)$" denotes the structure $(L_\alpha; \in)$ augmented by a predicate for $L_\beta$.

My question is:

Is $E_\kappa^+$ always a proper subset of $E_\kappa$?

*Note: This is a corrected version of Fine structure question: when do levels of $L$ look "a lot" like each other?, in which $E_\kappa^+$ was defined incorrectly (it was late and I was tired). I suspect the answer is the same - that $E_\kappa^+$ is much, much smaller than $E_\kappa$ - but the situation is significantly different: for one thing, Joel's answer to the lined question shows that $E_\kappa^+$ as defined there is in fact empty, which it is definitely not here.*

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