The isolation-by-distance central limit conjecture under varying population size

Let $t\geq 0$ and let $\psi_1,\psi_2\in\mathcal{D}(\mathbb{T}^d)$. Write $\eta_N$ for the scaling factor, $P_t^N(\psi_1,\psi_2)$ for the isolation-by-distance quantity, $Z_t$ for the limiting fluctuation field, $n_t$ for the limiting population-density field, and $\mathds{1}_{\diagdown}$ for the indicator appearing in the pairing. Isolation-by-distance central limit conjecture. For any $t\geq 0$ and any $\psi_1,\psi_2\in\mathcal{D}(\mathbb{T}^d)$, one has

$$\lim_{N\to\infty}N\eta_NP_t^N(\psi_1,\psi_2)=\mathbb{E}\left[\frac{\langle Z_t\otimes Z_t,(\psi_1\otimes\psi_2)\mathds{1}_{\diagdown}\rangle}{\langle n_t,\psi_1\rangle\langle n_t,\psi_2\rangle}\right].$$

When the population size is constant, the corresponding limit can be obtained using dominated convergence. For varying population size, the authors explain that a technicality prevents this argument under their current formulation; thus the conjectured limit remains unresolved in the stated generality.