Giraud's theorem for Segal topoi

Let $A$ be a $\mathbb{V}$-small Segal category. A $\mathbb{U}$-Segal topos is a Segal category satisfying the conditions in the claim below; a coproduct is disjoint when its canonical squares with the initial object are cartesian, and a groupoid object is a simplicial object satisfying the Segal and invertibility conditions. For a groupoid object $X_{*}$, write $|X_{*}|$ for the colimit of its simplicial diagram and $N(X_{0} \rightarrow |X_{*}|)$ for the nerve of the natural morphism.

Giraud's theorem for Segal topoi. $A$ is a $\mathbb{U}$-Segal topos if and only if:

1. The simplicial sets of morphisms in $A$ are essentially $\mathbb{U}$-small.
2. $A$ has all $\mathbb{U}$-small colimits, and coproducts in $A$ are disjoint.
3. For every groupoid object $X_{*}$ in $A$, the natural morphism

$$X_{*} \longrightarrow N(X_{0} \rightarrow |X_{*}|)$$

is an equivalence of simplicial objects in $A$. 4. Colimits in $A$ are stable by pullbacks: for every $\mathbb{U}$-small category $I$, every $I$-diagram $x_{*}$ in $A$, and morphisms from $\operatorname{Colim}_{i\in I}x_{i}$ to $z$ and from $y$ to $z$, the natural morphism

$$\operatorname{Colim}_{i\in I}\left(x_{i}\times_{z}y\right) \longrightarrow \left(\operatorname{Colim}_{i\in I}x_{i}\right)\times_{z}y$$

is an equivalence. 5. $A$ has a $\mathbb{U}$-small set of strong generators.

This is presented as the Segal-category analogue of Giraud's theorem, characterizing Segal topoi by smallness, colimits, disjoint coproducts, effective groupoid objects, pullback-stable colimits, and strong generators. The parser supplies no evidence that the claim has been resolved; its status is therefore open.