It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a presheaf $\infty$-category $\mathcal{P}r(\mathcal{A})$, for some small $\infty$-category $\mathcal{A}$, we can present $\mathcal{A}$ by a small simplicially enriched category $\mathbf{A}$, let $\mathbf{E} = \mathbf{sPr}(\mathbf{A})$ be simplicial presheaves on $\mathbf{A}$ with the projective or injective model structure, which presents $\mathcal{P}r(\mathcal{A})$, and left Bousfield localize it to present $\mathcal{E}$.
It's also known that any adjunction $f^* : \mathcal{E} \rightleftarrows \mathcal{F} : f_*$ between locally presentable $\infty$-categories can be presented by a Quillen adjunction between Quillen model categories that are themselves locally presentable (combinatorial). For instance, if we present them as left Bousfield localizations of the projective model structures on $\mathbf{sPr}(\mathbf{A})$ and $\mathbf{sPr}(\mathbf{B})$, then $f^*$ restricted to $\mathcal{A}$ is a small diagram in $\mathcal{F}$ and thus can be rectified to a simplicial functor $\mathbf{A} \to \mathbf{sPr}(\mathbf{B})$, and then we extend it cocontinuously to $\mathbf{sPr}(\mathbf{A})$ and descend it to the localization.
Now suppose $f^* : \mathcal{E} \rightleftarrows \mathcal{F} : f_*$ is a geometric morphism of $\infty$-topoi, so that $f^*$ preserves finite $\infty$-limits. Can this be presented by a Quillen adjunction between Quillen model structures on 1-toposes that is itself a 1-geometric morphism, i.e. whose Quillen left adjoint preserves finite strict 1-limits?
If not possible in general, I would be interested in any conditions on $\mathcal{E}$, $\mathcal{F}$, and $f$ that make it possible. The only sufficient condition I know of so far is that $\mathcal{E}$ and $\mathcal{F}$ are weakly 1-localic, i.e. left exact localizations of $\infty$-presheaves on some 1-category.
