Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\mathcal{J}\approx\mathcal{K}$" means that the logics $\mathcal{J}$ and $\mathcal{K}$ are equivalent in the sense that for every $\mathcal{J}$-sentence $\varphi$ there is a $\mathcal{K}$-sentence $\psi$ with the same model class and conversely.
I'm interested in two (types of) transformations of logics:
First, we have the various infinitarizations. Given a regular cardinal $\kappa$ and a logic $\mathcal{J}$, let $\mathbb{I}_\kappa(\mathcal{J})$ be the smallest logic containing $\mathcal{J}$ and closed under Boolean combinations of size $<\kappa$ and quantification over tuples of size $<\kappa$. And, given a singular cardinal $\lambda$, let $\mathbb{I}_\lambda(\mathcal{J})=\bigcup_{cf(\kappa)=\kappa<\lambda}\mathbb{I}_\kappa(\mathcal{J})$. Note that we have $\mathbb{I}_{\kappa_0}\circ\mathbb{I}_{\kappa_1}=\mathbb{I}_{\max\{\kappa_0,\kappa_1\}}$.
Second, we have modalization. Given a logic $\mathcal{J}$, let $\mathbb{M}(\mathcal{J})$ be the smallest logic $\mathcal{K}$ such that $\mathcal{J}\subseteq\mathcal{K}$ and, for every $\mathcal{K}$-sentence $\varphi$, there is a $\mathcal{K}$-sentence $\Box\varphi$ with the property that for every structure $\mathfrak{A}$ we have $\mathfrak{A}\models_\mathcal{K}\Box\varphi$ iff $\mathfrak{B}\models_\mathcal{K}\varphi$ for every $\mathfrak{B}$ containing $\mathfrak{A}$ as a substructure.
By an argument$^1$ similar to the proof of the Los-Tarski theorem, we have $\mathbb{M}(\mathbb{I}_{\theta}(\mathsf{FOL}))\approx\mathbb{I}_{\theta}(\mathsf{FOL})$ whenever $\theta$ is a limit of strongly compact cardinals, giving a conditional answer to Questions 57/58 of Hamkins/Woloszyn. Moreover, assuming Vopenka's Principle the same argument can be extended to arbitrary logics, yielding the following:
$(\star)$ For every logic $\mathcal{J}$ and every infinite cardinal $\kappa$ there is a logic $\mathcal{K}$ such that $\mathcal{J}\le\mathcal{K}$ and $$\mathcal{K}\approx\mathbb{M}(\mathcal{K})\approx\mathbb{I}_\kappa(\mathcal{K}).$$
This uses Makowsky's theorem that VP implies the existence of strong compactness numbers for all logics. Moreover, there is some heuristic evidence (namely this MO answer of Trevor Wilson) that we can't replace VP with anything weaker unless we find a genuinely different line of attack. My question is whether this is in fact the case:
Over the base theory $\mathsf{MK}$, does $(\star)$ have any nontrivial consistency strength?
Note that we require all logics to have $V$-definable semantics. This means that operators on logics such as $\mathbb{M}\circ\mathbb{I}_\kappa$ (say) do not automatically have fixed points; for example, see the second-to-last section of the above-linked Hamkins/Woloszyn paper for an instance of a seemingly-$\mathbb{M}$-like operator on logics which doesn't even have a fixed point above $\mathsf{FOL}$.
$^1$For completeness, let me sketch the argument mentioned above; I can add details if anyone is interested.
Suppose $\theta$ is a limit of strongly compact cardinals and $\varphi\in\mathcal{L}_{\theta,\theta}$; I'll show that $\Box\varphi$ is semantically equivalent to an $\mathcal{L}_{\theta,\theta}$-sentence (the rest of the argument is a routine induction on complexity, and a rephrasing of this paragraph to be about formulas rather than sentences). Let $\kappa$ be a strongly compact cardinal such that $\theta\in\mathcal{L}_{\kappa,\kappa}$, and let $\mathbb{S}$ be the set of $\mathcal{L}_{\kappa,\kappa}$-sentences which are purely existential and inconsistent with $\neg\varphi$. The disjunction $\sigma:=\bigvee\mathbb{S}$ is again an $\mathcal{L}_{\theta,\theta}$-sentence; I claim $\sigma$ is semantically equivalent to $\Box\varphi$. Trivially $\mathfrak{A}\models\sigma\implies\mathfrak{A}\models\Box\varphi$. In the other direction, suppose $\mathfrak{B}\models\Box\varphi$. Then the atomic diagram of $\mathfrak{B}$ is inconsistent with $\neg\varphi$. By strong compactness of $\kappa$, this means that there is a subset of the atomic diagram of $\mathfrak{B}$ of cardinality $<\kappa$ which is inconsistent with $\neg\varphi$. But this means $\mathfrak{B}\models\sigma$.
