Functoriality of the lasso construction on schemata

A schema is a small category, and for a schema $\mathsf{T}$, $\mathsf{Set}^{\mathsf{T}}$ denotes its associated copresheaf category. Let $\mathsf{Lasso}(\mathsf{Set}^{\mathsf{T}})$ be the category of lassos on that copresheaf category. Lasso functoriality conjecture. There is a contravariant functor

$$\mathsf{Lasso} \colon \mathsf{Cat} \to \mathsf{Cat}$$

which sends every functor $F \colon \mathsf{S} \to \mathsf{T}$ between schemata to a functor

$$\mathsf{Lasso}(\mathsf{Set}^{\mathsf{T}}) \to \mathsf{Lasso}(\mathsf{Set}^{\mathsf{S}})$$

between the corresponding categories of lassos. This would establish a systematic relationship between a schema and the category of lassos on its copresheaf category; the source presents it as a conjecture and does not provide a resolution.