Continuum-limit conjecture for discretized higher gauge connections

Let $M$ be a manifold and let ${\cal D}$ range over all discretizations of $M$. Denote by $\boldsymbol{\operatorname{Conn}} /\!/ \boldsymbol{\operatorname{Gauge}}(M,\mathcal{G})$ the continuum double groupoid of connections and gauge transformations, and by $\boldsymbol{\operatorname{Conn}} /\!/ \boldsymbol{\operatorname{Gauge}}(M,\mathcal{D},\mathcal{G})$ the corresponding double groupoid for the discretization ${\cal D}$. Continuum-limit conjecture. The continuum double groupoid is the inductive limit

$$\boldsymbol{\operatorname{Conn}} /\!/ \boldsymbol{\operatorname{Gauge}}(M,\mathcal{G})$$

of the discretized double groupoids

$$\boldsymbol{\operatorname{Conn}} /\!/ \boldsymbol{\operatorname{Gauge}}(M,\mathcal{D},\mathcal{G})$$

over all discretizations ${\cal D}$ of $M$. This proposes a way to recover the continuum theory from increasingly fine discretizations, ordered by refinement, although the relevant analytic and categorical foundations are not supplied here.