Is there an abstract logic that defines the mantle?

It is a known result by Scott and Myhill that the second-order version of $L$ yields $mathrm{HOD}$.
Recently, Kennedy, Magidor, and Väänänen (Inner models from extended logics: Part I and II) investigated inner models given by logics with generalized quantifiers, which yields a logic intermediate between first-order and second-order logic. It motivates the following question:

Is there a logic that produces the mantle?

(Here the choice of the mantle is somewhat arbitrary; we may replace it by ‘generic mantle’, ‘symmetric mantle’ or whatever. I will focus on the mantle in this question, but I welcome discussing other cases.)

Of course, the answer is trivial if we assume, like $V=L$ or $V=L[G]$ for some $L$-generic $G$. I want to ask the existence of logic which defines the mantle uniform to models of ZFC.

Is there a (ZFC-definable) abstract logic $mathcal{L}$ such that the inner model given by $mathcal{L}$ is (ZFC-provably) the mantle?

(Under model-theoretic terms, is there $mathcal{L}$ such that for any model $M$ of $mathsf{ZFC}$, the inner model given by $mathcal{L}$ is the mantle of $M$?)

Here are some of my rough thoughts:

• Sublogics of higher-order logics are not the candidate for $mathcal{L}$: the corresponding inner models of higher-order logics are $mathrm{HOD}$ (if my reasoning is correct), so the sublogics yield a submodel of $mathrm{HOD}$. However, $mathrm{HOD}$ need not be the mantle. (Theorem 70 of Fuchs, Hamkins, and Reitz (Set-theoretic geology).)

• We can rule out $mathcal{L}_{kappakappa}$, which yields Chang model. The inner model given by $mathcal{L}_{kappakappa}$ is the least transitive model of ZF that contains all ordinals and is closed under $<kappa$-sequences (Theorem II of Chang’s Sets constructible using $L_{kappakappa}$.) However, the mantle need not be closed under $<kappa$-sequences. (A generic extension of $L$ would be an example.)

I would appreciate any comments or answers.