The finiteness conjecture for bases without interesting reversed sum-product pairs

Let $\beta\geq 2$ be a base. A pair $(a,b)$ is a reversed sum-product pair in base $\beta$ when the sum of $a$ and $b$, written in base $\beta$, is the reversal of the base-$\beta$ representation of their product. The pairs $(2,2)$ and $(\beta-1,\beta-1)$ are called uninteresting; every other reversed sum-product pair is interesting. The finiteness conjecture. The only bases for which there is no interesting reversed sum-product pair $(a,b)$ are

$$2,3,4,5,6,7,8,9,12,15,21.$$

The paper presents this as an intriguing phenomenon that the authors are unable to explain; the stated list is therefore conjectural rather than established by the preceding finiteness theorem for each fixed base.