If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\mathbf{H}/X$. Then the category $T \mathbf{H}$ is an $\infty$-topos (see this page for a proof). The question is: can this fact be generalized to other Cartesian fibrations constructed in a similar way? For example, if we have an essentially algebraic $\infty$-theory $S$, then we can define in a similar way a Cartesian fibration $p_S : T_S \mathbf{H} \to \mathbf{H}$ such that the fiber of $p_S$ over $X$ is the $\infty$-category of models of $S$. Is it true that $T_S \mathbf{H}$ is an $\infty$-topos?

If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\mathbf{H}/X$. Then the category $T \mathbf{H}$ is an $\infty$-topos (see this page for a proof). The question is: can this fact be generalized to other Cartesian fibrations constructed in a similar way? For example, if we have an essentially algebraic $\infty$-theory $S$, then we can define in a similar way a Cartesian fibration $p_S : T_S \mathbf{H} \to \mathbf{H}$ such that the fiber of $p_S$ over $X$ is the $\infty$-category of models of $S$ in $\mathbf{H}/X$. Is it true that $T_S \mathbf{H}$ is an $\infty$-topos?