Gaussian cycle-count conjecture for permutations with averaged cycle weights

Let $\vec{\sigma}=(\sigma_1,\sigma_2,\ldots)$ be a sequence of nonnegative real numbers with mean $\alpha$, meaning that

$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\sigma_k=\alpha.$$

Permutations of $[n]$ selected according to the weights $\vec{\sigma}$ should have an asymptotically Gaussian number of cycles as $n\to\infty$, with mean and variance asymptotic to $\alpha\log n$. This conjecture predicts that the average of the cycle weights controls the limiting cycle-count behaviour, extending analogous results for Boltzmannized permutations; the stated context gives supporting evidence but does not establish the conjecture for the general sequences covered here.