Conjecture on the occurrence of every digit in algebraic irrational expansions

Let $x$ be a real irrational algebraic number, let $g\geqslant3$ be a positive integer, and let $a$ be an integer satisfying $0\leqslant a\leqslant g-1$. The $g$-ary expansion of $x$ is the expansion of $x$ in base $g$, whose digits belong to $\{0,\ldots,g-1\}$. Digit-occurrence conjecture. The digit $a$ occurs at least once in the $g$-ary expansion of $x$. This is a very weak expected property of expansions of irrational algebraic numbers; the source notes that even explicit examples for which a specified digit occurs infinitely often are not known.