Limiting average class size ratio conjecture

For character tables included up to size $N$, let $F_{\mathrm{cls}}(N,l)$ denote the cumulative ratio associated with the average conjugacy class size of class-algebra generating sets, through $n=l$. Assuming the limit exists, define

$$L(l)=\lim_{N\to\infty}F_{\mathrm{cls}}(N,l).$$

Average class size conjecture (weak version (b)). The limiting ratio satisfies

$$\lim_{N\to\infty}F_{\mathrm{cls}}(N,l)\geq1$$

for every $l$.

This is the second part of the weaker large-$N$ formulation and is posed in the source as an open question. It expresses the expectation that the limiting cumulative ratio retains the strong conjecture's inequality.