TITLE: Elementary submodels in stationary logic.
QUESTION [7 upvotes]: In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of size at most $\aleph_1$. (only countable signatures are considered)
It is then asked whether in fact the stronger result holds, that every structure has an elementary substructure of size at most $\aleph_1$.
In a footnote on p.222 it is mentioned that this question depends on set theory:
"If V = L, then there is a structure of size $\omega_2$ with no L(aa)-elementary submodel of power $\omega_1$"
I'm looking for a reference where this argument is worked out in more detail or if it is not too long maybe someone here can give me a more detailed argument.
Edit: L(aa) arises from first order logic in the following way:
We add new variables $X_1,X_2,\dots$ for unary countable relations, which lead to new atomic formulas $Xt$ for a first order term t. Now the formation rules for formulas are that of first order logic together with the new quantifier (aa) which binds one relation variable, i.e if $\varphi$ is a formula, so is $(aa)X_i\varphi$ the meaning of which is defined by
$$\mathfrak{A}\models(aa)X_i\varphi \Leftrightarrow \{R\in P_{\omega_1}(A)|\mathfrak{A}\models \varphi[R]\}\in D(A)$$
where $D(A)$ is the club filter on $P_{\omega_1}(A)$. So aa says for almost all relations.

REPLY [5 votes]: The assertion that ``the L-S number of stationary logic is $\omega_2$" is equivalent to a principle I introduced a while back, called "Diagonal Reflection Principle for internally club sets" ($\text{DRP}_{\text{IC}}$).  The equivalence is proved in the following recent preprint, available on Fuchino's website:
--Sakaé Fuchino, André Ottenbreit Maschio Rodrigues and Hiroshi Sakai, Strong Löwenheim-Skolem theorems for stationary logics, I
This principle is very strong, since it implies failures of many square principles (and is expected to be equiconsistent with a supercompact).  In particular, it fails in L.