Largest cyclic critical group conjecture for arithmetical structures on star and complete graphs

Let $a_1=1$ and $a_n=a_{n-1}^2+a_{n-1}$ for $n\geq 2$. For an arithmetical structure on the star graph $S_n$ or the complete graph $K_n$, its critical group is a finite abelian group; a cyclic critical group is one isomorphic to $\mathbb{Z}/m\mathbb{Z}$ for some positive integer $m$. The construction considered has critical group $\mathbb{Z}/a_{n-1}\mathbb{Z}$. Largest cyclic critical group conjecture. For $n\geq 4$, the largest cyclic group that can be realized as a critical group of an arithmetical structure on $S_n$ or on $K_n$ is

$$\mathbb{Z}/a_{n-1}\mathbb{Z}.$$

The preceding construction realizes this cyclic critical group using the Sylvester-sequence arithmetical structure. The conjecture asserts that no arithmetical structure on either graph realizes a cyclic critical group of larger order for $n\geq 4$.