Conjecture on the thermodynamic limit of the effective reduced density-matrix series

Let $\rho_N(x,t)$ be the finite-volume one-particle reduced density matrix. Let $w_L\sim L^{1+1/4}$, define

$$\mathcal{B}_L=\{n\in\mathbb{Z}:-w_L<n<w_L\},\qquad \mathcal{B}_L^{\mathrm{ext}}=\mathcal{B}_L\setminus\{1,\ldots,N+1\},\qquad \mathcal{B}_L^{\mathrm{int}}=\{1,\ldots,N+1\},$$

and let $\mu_{p_a}$ and $\mu_{h_a}$ be the rapidities associated with particle indices $p_a\in\mathcal{B}_L^{\mathrm{ext}}$ and hole indices $h_a\in\mathcal{B}_L^{\mathrm{int}}$. Let $\widehat D_N\widehat{\mathcal G}_{N;1}$ and $F_0$ be the functionals and form-factor data appearing in the effective series.

Effective-series thermodynamic-limit conjecture.

$$\lim_{N,L\to+\infty}\rho_N(x,t)=\lim_{N,L\to+\infty}\rho_{N;\mathrm{eff}}(x,t),$$

where

$$\rho_{N;\mathrm{eff}}(x,t)=\sum_{n=0}^{N+1}\sum_{\substack{p_1<\dots<p_n\p_a\in\mathcal{B}_L^{\mathrm{ext}}}}\sum_{\substack{h_1<\dots<h_n\h_a\in\mathcal{B}_L^{\mathrm{int}}}}\prod_{a=1}^{n}\frac{e^{-\mathrm{i}xu(\mu_{h_a})}}{e^{-\mathrm{i}xu(\mu_{p_a})}}\cdot(\widehat D_N\widehat{\mathcal G}_{N;1})\left(\begin{array}{c}\{p_a\}_{1}^{n}\\\{h_a\}_{1}^{n}\end{array}\right)\left[F_0\left(*\left|\begin{array}{c}\{\mu_{p_a}\}\\\{\mu_{h_a}\}\end{array}\right.\right);\xi;\xi_{F_0}\right].$$

Here the star denotes the running variable of $F_0$ on which the two functionals act.

The conjecture formalizes the approximation obtained by restricting excitation integers, dropping the stated finite-volume remainders, and replacing the oscillating exponent by its thermodynamic limit. The supplied text does not establish the equality, so its resolution remains open.