Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$. Letting $i=\{i_\Sigma:\Sigma$ a finite language$\}$, we can define an inner model $$M_\mathcal{L}^r:=L[\{\langle\mathfrak{A},n\rangle: \mathfrak{A}\models_\mathcal{L}i_{\mathit{Lang}(\mathfrak{A})}^{-1}(n)\},r]$$ for each real $r$. Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:
Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$
Due to the on-a-cone focus, the specific $i$ used will play no role.
Intuitively, the $M_\mathcal{L}^r$s represent a very naive attempt to generalize the usual construction of $L$. I'm curious how much of the usual theory of $L$ is retained. A previous question asked about large cardinals; here, I'll focus on combinatorics. Specifically, I'm interested in the extent to which models of the form $M_\mathcal{L}^r$, for $r$ "strong enough" relative to $\mathcal{L}$, have some analogue of fine structure. Since such models automatically have some decent condensation properties, this is unfortunately a bit technical - I need to focus on some concrete test question which doesn't just follow from condensation alone.
The best candidate I know right now is $\Box_{\omega_1}$, for the following reason. Woodin's Axiom of Strong Condensation, introduced in his nonstationary ideal book and further analyzed in Law's thesis An abstract condensation property, provides a framework for "coarse" (as opposed to fine) structure theory. Wu's paper Set forcing and strong condensation for $H(\omega_2)$ established a concrete limitation on what this sort of coarse structure theory can do by showing that $\mathsf{ZFC+ASC\not\vdash\Box_{\omega_1}}$. So based on this, my question is:
Is there a good logic $\mathcal{L}$ such that $\Box_{\omega_1}\not\in \mathsf{IIMT}_\mathcal{L}$?
