Functoriality conjecture for the span construction

Let $(\infty,2)$-categories have categories as objects, functors as 1-morphisms, and natural transformations as 2-morphisms. Consider the $(\infty,2)$-category whose objects are categories with pullbacks, whose 1-morphisms are functors preserving pullbacks, and whose 2-morphisms are cartesian natural transformations. For a category $\mathcal C$ with pullbacks, let $\operatorname{Span}(\mathcal C)$ denote its category of spans.

Span functoriality conjecture. The $\operatorname{Span}$ construction underlies a limit-preserving $(\infty,2)$-functor from the $(\infty,2)$-category of categories with pullbacks, functors preserving pullbacks, and cartesian natural transformations to the $(\infty,2)$-category of categories.

The results establishing functoriality, composition of natural transformations, and extension of natural transformations provide a $2$-categorical approximation to this conjecture, together with the equivalence $\operatorname{Span}(\mathcal C\times\mathcal D)\equiv\operatorname{Span}(\mathcal C)\times\operatorname{Span}(\mathcal D)$. The conjectured higher-categorical, limit-preserving functoriality remains to be established.