Conjecture on the limiting measure and skew-adjoint loop operators

Let $u_\kappa^\sigma$ be the measures from the existence construction, let $\gamma=g\hat g^{-1}$ be sampled from $\nu_\kappa^\sigma$, and let $\hat\nu_\kappa^\sigma$ be the law of $\hat g^{-1}\partial\hat g$. Set

\hat\mathcal A=\{\alpha^*\mid\alpha\in\mathcal A\}.

The operators $\mathbf R_u$ are indexed by $u\in L^\omega\mathfrak g_\mathbb C$.

Limiting-measure and operator conjecture. The family $\hat\nu_\kappa^\sigma$ converges weakly as $\sigma\to0$ to a measure $\hat\nu_\kappa$ on \hat\mathcal A. Moreover, the operators $(\mathbf R_u)_{u\in L^\omega\mathfrak g_\mathbb C}$ are densely defined and closable on $L^2(\hat\nu_\kappa)$ and satisfy

$$\mathbf R_u^*=-\mathbf R_{u^*}.$$

This conjecture addresses the analytic difficulty caused by inverse loops and the failure of the corresponding formal adjoint relation for the original measure. Its resolution is not established in the supplied text.