Asymptotic normality of the number of independent-set classes in forests

Let $(F^{c(n)}_n)_{n \geq 0}$ be a sequence of forests, where $F^{c(n)}_n$ has $n$ vertices and $c(n)$ components. Let $X^{c(n)}_n$ be the number of classes in a uniformly chosen partition of the vertex set of $F^{c(n)}_n$ into non-empty independent sets. Asymptotic-normality conjecture. The sequence $(X^{c(n)}_n)_{n \geq 0}$ is asymptotically normal for all $1 \leq c(n) \leq n$. The theorem preceding this conjectural extension establishes asymptotic normality when $c(n)<C\sqrt{n/\log n}$ for some constant $C>0$; the conjecture proposes the result for the full range of possible component counts.