Conjectured existence of complexifier-based semi-classical states

Let $M$ be the underlying 3-manifold, let $\mathbf{HD}(M)$ be the holonomy-diffeomorphism algebra, and let $fF\in\mathbf{HD}(M)$ have corresponding approximations $\mathbb{fF}_n$. Let $\phi_n^t$ be semi-classical states associated with a complexifier, let $\varphi$ denote the representation, and let $\varphi_A$ denote evaluation at a classical connection $A$ in the spinor state $v$. For a sequence of vector fields $\mathcal{L}_{{\bf e}^a_{i_n}}$ associated with graphs $\Gamma_n$ and edges converging to $x$, define the limiting electric-field observable by the expression in the requirements below.

Complexifier conjecture. There exists a choice of complexifier giving a sequence $\phi^t=\{\phi_n^t\}$ of semi-classical states satisfying

$$\mathcal{E}(fF):=\lim_{n\to\infty}\langle\phi_n^t\mid\varphi(\mathbb{fF}_n)\mid\phi_n^t\rangle<\infty,$$ $$\mathcal{E}(\hat E_m^a(x)):=\lim_{n\to\infty}\langle\phi_n^t\mid t2^{2n}\mathcal{L}_{{\bf e}^a_{i_n}}^n\mid\phi_n^t\rangle<\infty,$$ $$\lim_{t\to0}\mathcal{E}(fF)=(v,\varphi_A(fF)v),\qquad \lim_{t\to0}\mathcal{E}(\hat E_a^m(x))=\mathrm{i}E_a^m(x).$$

These conditions encode the desired semiclassical limits for holonomy-diffeomorphism and electric-field observables. The paper states the existence as a conjecture and does not resolve it.