Local asymptotic conjecture for generalized abundancy coefficients

Let $cmathcal{H}_{\ell,n}(x)=\sum_{k=0}^{n}A(\ell,n,k)x^k/n!$, and let $\lambda_{\ell}=\ell\ln(\ell-1)-\ln(\mathcal{K}_{\ell})$, where $\mathcal{K}_{\ell}=\Gamma(\ell)\zeta(2)\cdots\zeta(\ell)$. For $n\to\infty$ and $k\to\infty$ such that $k/n^{(\ell-1)/\ell}$ converges to a quantity in $(0,\infty)$, define $\widetilde{\lambda}_{\ell}(k/n)=\lambda_{\ell}+\varepsilon_{\ell}(k/n)$, where $\varepsilon_{\ell}(t)$ is a linear combination of terms $t^j(\ln t)^r$ with $1\leq j\leq\ell-1$ and $r\in\{0,1,\dots\}$. Main conjecture. Under these conditions, we should have

$$\frac{A(\ell,n,k)}{n!}=\frac{\sqrt{\ell-1}}{2\pi n}e^{k\ell-k\ell\ln(k)+(\ell-1)k\ln(n)-\widetilde{\lambda}_{\ell}(k/n)k}(1+o(1)).$$

The conjecture refines the preceding central limit theorem by incorporating corrections to $\lambda_{\ell}$ that are needed to obtain order-one accuracy. The paper notes that such corrections can be calculated for $\mathcal{H}_{\ell,n}(1)$ in related work, but does not establish this local asymptotic formula.