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Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that theL\"owenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent tothe Diagonal Reflection Principle on internally club sets($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that therestriction of the LST property to (downward) reflection of $\Pi^1_1$ formulas,which we call the $\Pi^1_1 \! \! \downarrow$-LST property, is equivalent to the\emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with resultsfrom \cite{Cox_RP_IS}, this shows that the $\Pi^1_1 \! \! \downarrow$-LSTProperty for Stationary Logic is strictly weaker than the full LST Property forStationary Logic, though if CH holds they are equivalent.

The $\Pi^1_1 \! \! \downarrow$ L\"owenheim-Skolem-Tarski property of Stationary Logic