Span preserves lax-univalent discrete cocartesian fibrations

Let $\boxed{}$ be an algebraic pattern and let $\boxed{}$ be a univalent discrete cocartesian fibration. The construction $\boxed{}$ sends it to a lax-univalent discrete cocartesian fibration internally to $\boxed{}$:

Span-preservation conjecture. Let $\boxed{}$ be a univalent discrete cocartesian fibration. Then

$$\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}_{\bullet})\to\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G})$$

is a lax-univalent discrete cocartesian fibration internally to $\boxed{}$.

This conjecture extends the preceding result for algebraic patterns whose morphisms are all inert to patterns with non-trivial active morphisms. The paper explains that lax morphisms, rather than only strong morphisms, are the relevant maps in this setting; the conjecture concerns the resulting classification of discrete cocartesian fibrations.