Asymptotic sparsity of composites in finite-group semirings

Let $G$ be a finite group, let $\mathbb{N}[G]_n$ denote the $n$-th grade of the group semiring, and let $\operatorname{Comp}(\mathbb{N}[G])_n$ denote its composite elements in that grade. Let $P^-(n)$ be the smallest prime factor of $n$, and suppose that the number of prime factors of $n$ is bounded by a fixed positive integer $k$. Asymptotic sparsity conjecture. One has

$$\lim_{P^-(n)\rightarrow\infty}\frac{\left|\operatorname{Comp}\left(\mathbb{N}[G]\right)_n\right|}{\left|\mathbb{N}[G]_n\right|}=0.$$

The theorem preceding this conjecture supplies an upper bound for composites and suggests that, under the bounded-prime-factor condition, composites are much less numerous than primes in the $n$-th grade. The conjecture itself is not resolved in the supplied text.