Asymptotic formula for suffix-restricted squares in base 10

For a base $B\ge 2$ and a digit set $D\subseteq\{0,1,\dots,B-1\}$ with $|D|\ge 2$, let $\omega_B^D(n)$ denote the number of $n$-digit suffixes in base $B$ from $D$ that occur as the suffix of a square. Suffix-count conjecture. For all $n\ge 3$,

$$\omega_{10}^{\{7,8,9\}}(n)=8\cdot 3^{n-3}.$$

This is a computer-supported conjectural pattern for the specified suffix digits; the source gives no proof or resolution.