In model theory, the Keisler-Shelah isomorphism theorem asserts that two models of a theory are elementary equivalent if and only if they have isomorphic ultrapowers. On the other hand, assuming that $\kappa$ is strongly compact, we can ensure the existence of a $\kappa$-complete ultrafilter on every set (that is, an ultrafilter closed under intersections of less than $\kappa$ elements) and prove a version of Łoś theorem for the infinitary logic $\mathcal{L}_{\kappa, \kappa}$.

My question is whether a suitable version of the Keisler- Shelah theorem also holds for infinitary languages. Shelah's original proof (the first without using GCH), or the similar version in Chang-Keisler book "Model theory", seem to be readily generalizable, but I would like if possible to find a reference for it. The statement I have in mind is as follows:

Let $\kappa$ be a strongly compact cardinal and assume $\mathcal{A}$ and $\mathcal{B}$ are models of a theory in $\mathcal{L}_{\kappa, \kappa}$. Then $\mathcal{A}$ and $\mathcal{B}$ are elementary equivalent (with respect to the language $\mathcal{L}_{\kappa, \kappa}$) if and only there is a set $I$ and a $\kappa$-complete ultrafilter $U$ over $I$ such that the ultrapowers $\Pi_{i \in I}\mathcal{A}/U$ and $\Pi_{i \in I}\mathcal{B}/U$ are isomorphic.

Does this hold or has it been considered in the literature?