The bad-part density limit conjecture for cyclic and dihedral groups

Let $BP(G)$ be the set of bad parts of a finite group $G$, let $\alpha(G)=\frac{|BP(G)|}{2^{\kappa(G)-1}-1}$, and let $p_n$ be the $n$-th prime number. Define

$$\beta_n=\alpha(Z_{p_n}),\qquad \gamma_n=\alpha(D_{2p_n}).$$

Bad-part density limit conjecture.

$$\lim_{n\rightarrow\infty}\beta_n=\lim_{n\rightarrow\infty}\gamma_n=1.$$

The conjecture is suggested by computations for cyclic and dihedral groups, asserting that the proportion of bad parts tends to $1$ along these prime-indexed families. The supplied text does not establish a proof or disproof; it later notes that analogous calculations for groups of order $3p$ do not support the conjecture in that broader setting.