Finiteness conjecture for digit-special integers

A real number $x\geq 0$ is digit-special if, for every integer base $b\geq 3$, its base-$b$ expansion omits at least one digit from $0,1,\dots,b-1$. Finiteness conjecture. There are finitely many digit-special integers. The paper presents this as a strengthening of its conditional theorem under Schanuel's conjecture; it remains open.