Span constructions provide enough lax-univalent fibrations

Let $\boxed{}$ be a sufficient family of univalent fibrations for $\boxed{}$, indexed by $\boxed{}$. Consider the corresponding family obtained by applying the span construction.

Sufficiency conjecture. Suppose $\boxed{}$ is a sufficient family of univalent fibrations for $\boxed{}$. Then

$$\left(\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}^{(\kappa)}_{\bullet})\to\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}^{(\kappa)})\right)_{\kappa\in K}$$

provides enough lax-univalent fibrations for $\mathfrak{Cat}(\mathfrak{Seg}_{\mathfrak{P}}(\mathfrak{\infty\textnormal{-}Grpd}))$.

The conjecture asserts that every relevant class of discrete cocartesian fibrations in the Segal-object setting is classified by a universal fibration obtained from a sufficient family in the ambient $(\infty,1)$-categorical setting. It complements the preceding conjecture, which identifies the lax-univalence of each span-constructed fibration.