Waring's problem with restricted digits

Let $k\ge 2$ and $b\ge 3$ be integers, and suppose that $d_1,d_2\in \{0,1,\dots ,b-1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then every sufficiently large integer is a sum of at most $b^{160k^2}$ numbers of the form $x^k$, $x\in \mathcal{S}$.