Bourgain's short-interval circle congruence conjecture

Let $n$ be a positive integer and consider positive integer pairs $(a_i,b_i)$ on the circle

$$a_i^2+b_i^2=n.$$

Bourgain's circle congruence conjecture. There exist $\delta>0$ and an integer $m>0$ such that, if

$$a_i^2\equiv a_1^2\pmod q\qquad (i=1,\ldots,m),$$

then

$$q=O(n^{1-\delta}).$$

The source gives this as an open conjectural consequence concerning clustered lattice points on circles.