Vinogradov's theorem for primes with restricted digits

Let $g$ be sufficiently large, $b\in \{0,\dots ,g-1\}$, and ${\mathcal{S}}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1+p_2+p_3$ where $p_i$ are prime and $p_i\in {\mathcal{S}}_b$.