The size bound for the set of mutual divisibility solutions

Let

$$\mathcal{A}(n)=\{a:1\leq a<n,\ n\mid a^2-1,\ a\mid n^2-1\}.$$

Size-bound conjecture. For any integer $n>1$,

$$|\mathcal{A}(n)|\leq 3.$$

The bound is equivalent to the assertion that any two chains $\mathfrak{C}_{k_1}$ and $\mathfrak{C}_{k_2}$, with $k_1>k_2\geq 3$, do not share an integer. The source presents this as an open question motivated by the observed size property.