The folklore conjecture on integers with binary digits in bases 3, 4 and 5

For an integer, say that its base-$b$ expansion contains solely the digits $0$ and $1$ when no other digit occurs. Folklore base-expansion conjecture. The only integers whose base-$3$, base-$4$, and base-$5$ expansions contain solely the digits $0$ and $1$ are

$$0,1,82000.$$

This conjecture is presented as an intriguing folklore problem concerning a known integer sequence; the paper reports progress toward it, but does not resolve it.