The maximal-order sporadic unit-equation conjecture

Let $K_a$ be the simplest cubic field considered in the paper, let $\mathcal{O}_{K_a}$ be its maximal order, and let $u_1,u_2\in\mathcal{O}_{K_a}^*$ and $n\in\mathbb{Z}$ satisfy the unit equation

$$u_1+u_2=n.$$

Two solutions are called non-equivalent according to the equivalence relation used for the unit equation, and a solution is sporadic when it is not one of the parametrized non-sporadic solutions. The maximal-order unit-equation conjecture. The unit equation has exactly $66$ sporadic solutions that are pairwise non-equivalent. The conjecture extends the result from the order $\mathbb{Z}[\rho]$ to units of the maximal order; computations in the range $-1\leq a\leq 400$ and $1\leq n\leq 400$ found $66$ such solutions, all satisfying $a\leq 66$ and $|n|\leq 22$, but the unrestricted claim remains open.