The second-smallest-value conjecture for the function $f$ on finite groups

Let $f$ be the function on finite groups studied in the paper, with $f(G)=1$ precisely for cyclic groups. Second-smallest-value conjecture. $2$ is the second smallest value of the function $f$. This conjecture is motivated by the preceding lower bounds for non-cyclic groups and has been verified by computer for many classes of finite groups; its general validity remains open.