On the unit equation $\epsilon +\delta =n$ in cubic fields

Let $n$ be an integer not equal to $-2$, $0$ or $2$. We consider the unit equation $\epsilon +\delta =n$ in units $\epsilon ,\delta$ of cubic fields. We show that this equation has no solutions for 100% of cubic fields, when ordered by discriminant. This is consistent with a recent conjecture of the authors.