Dupuyがモデル理論の解釈を持ち出したら
Scholzeに「それ、あたってなくね？」と一蹴されたね

For all I can see, Hodge theaters fit neither into the framework of “structures” as used in the wikipedia entry
https://en.wikipedia.org/wiki/Interpretation_(model_theory)
you linked to, nor the topos-theoretic framework of Caramello.
(Regarding the first one: A “structure” in the sense of model theory has first of all an underlying set.
I find it hard to take a Hodge theater and produce some interesting set that is functorial in isomorphisms of Hodge theaters, the problem being the very lax notion of isomorphisms of Hodge theaters.)

However, these long discussions are all about interpretations.
Regarding the mathematics proper: I stand by the claim made in our　manuscript, and have indicated the proof above.