Critical non-standard central limit conjecture for the triangle count

Let $T_n$ be the triangle count under the edge-triangle model with law ${\mathbb P}_{n;\alpha,h}$, and let $(\alpha_c,h_c)$ be the critical parameter pair. At this pair, let $m_n^\Delta(\alpha_c,h_c)$ be the mean-field centering term. Non-standard central limit conjecture for the triangle count. If $(\alpha,h)=(\alpha_c,h_c)$, then

$$6\,\frac{\frac{T_n}{n}-\frac{n^2}{6}m_n^\Delta(\alpha_c,h_c)}{n^{3/2}}\xrightarrow{\mathrm d}Y$$

with respect to ${\mathbb P}_{n;\alpha_c,h_c}$ as $n\to+\infty$, where $Y$ is a generalized Gaussian random variable with Lebesgue density

$$\ell^c(y)\propto e^{-\frac{3^8}{2^{14}}y^4}.$$

This conjecture describes the quartic, non-Gaussian fluctuation expected at criticality. It is part of the proposed extension of the mean-field asymptotics to the edge-triangle model, whose asymptotic equivalence remains an open problem.