Rigorous Wilson loop observable existence conjecture

Let $L$ satisfy Assumption, and let $\operatorname{WLO}_{rig}(L)$ be the regularized Wilson loop observable defined by

$$\operatorname{WLO}_{rig}(L):= \lim_{n \to \infty} \lim_{s \to 0} \lim_{\epsilon\to 0} \sum_{y \in I} \int_{\sim} \Psi\bigl( J^{(s,\epsilon,n)}_{b,y}(L)\bigr)\, db.$$

Here $J^{(s,\epsilon,n)}_{b,y}(L)$ is the regularized integrand defined for $b \in \mathfrak t$, $y \in I$, $s \in (0,1)$, $\epsilon \in (0,\epsilon(s))$, and $n \in \mathbb N$, while $\int_{\sim}$ denotes the mean-value integral over $\mathfrak t$.

Existence conjecture. $\operatorname{WLO}_{rig}(L)$ is well-defined. In particular, all limits involved exist.

This conjecture asserts the existence of the regularized Chern–Simons Wilson loop observable obtained from the proposed infinite-dimensional path-integral construction. The source does not provide a resolution of the conjecture; its validity is used to motivate the subsequent comparison with the shadow invariant.