The sporadic unit-equation conjecture for simplest cubic orders

Let $K_a$ be the simplest cubic field considered in the paper, let $\rho$ generate the order $\mathbb{Z}[\rho]$, and let $u_1,u_2\in\mathbb{Z}[\rho]^*$ and $n\in\mathbb{Z}$. Two solutions are called non-equivalent according to the equivalence relation defined for the unit equation in the paper. The parameter $a$ is the parameter defining $K_a$. The sporadic unit-equation conjecture. The Diophantine equation

$$u_1+u_2=n$$

has exactly $24$ sporadic solutions that are pairwise non-equivalent. Moreover, every sporadic solution $(u_1,u_2,n)$ satisfies $a\leq 2$ and $|n|\leq 22$. This strengthens the theorem proved under the additional assumption $|n|\leq\max\{|a|^{1/3},1\}$; the computational evidence reported in the paper suggests that no further sporadic solutions occur for $a\geq 3$, but the unrestricted assertion remains conjectural.