The rational-points conjecture for sparse missing digits on the unit circle

Let $C$ be the unit circle

$$C:x^2+y^2=1.$$

For sufficiently large integers $p$, let $D\subset\{0,\dots,p-1\}^2$ satisfy $\#D\leq100$, and let $K_{p,D,l}$ denote the corresponding missing-digits set at level $l$. Rational-points conjecture. The intersection $C\cap K_{p,D,l}$ contains only rational points for every $l\geq0$. This is posed as a concrete problem about the scarcity of points with small missing-digits sets on a curved manifold. No resolution is supplied.