Nondegenerate scaling limits for Wick powers of the loop-soup field

Let $\theta \in (0,1/2)$, let $h_{\theta,N}$ be the discrete loop-soup field, and define its Wick powers by

$$:h_{\theta,N}(x) \ell_x^n: = h_{\theta,N}(x) G(x,x)^n \mathbf{L}_n^{(1-\theta)}(\ell_x/G(x,x)),$$

where $G(x,x)$ is the relevant Green function and $\mathbf{L}_n^{(1-\theta)}$ is the Laguerre polynomial. Let $\mathcal{L}_{D_N}^\theta$ be the discrete loop soup, and let $H^{-\varepsilon}$ and $\mathfrak{L}$ denote the spaces used for the field and loop-soup components. The Wick-power scaling-limit conjecture. For any $n_0\geq 1$, the joint law converges in distribution as

$$\left(h_{\theta,N}, (:h_{\theta,N} \ell^n:)_{n=1 \dots n_0}, \mathcal{L}_{D_N}^\theta\right) \xrightarrow[N\to\infty]{\mathrm{(d)}} \left(h_\theta^{\mathrm{conj}}, (:h_\theta^{\mathrm{conj}} L^n:)_{n=1 \dots n_0}, \mathcal{L}_D^\theta\right).$$

The convergence holds in $(H^{-\varepsilon})^{n_0}\times\mathfrak{L}$ for every $\varepsilon>0$, and the limiting fields are measurable with respect to $\mathcal{L}_D^\theta$. This conjecture proposes scaling limits for the renormalised odd powers not covered by the already defined integer local-time powers; the existence and nondegeneracy of these limits remain open.