The congruence-spacing conjecture for square roots modulo integers

For integers $a$ and $b$, consider positive solutions of $x^2\equiv a\pmod b$. The congruence-spacing conjecture. There exists a constant $N$ such that, for every $a$ and $b$, there do not exist more than $N$ solutions

$$0<x_1<x_2<\cdots<x_N<x_1+b^{1/2}$$

to this congruence. The source notes that this would imply Rudin's square-counting conjecture for arithmetic progressions, while only weaker logarithmic bounds are proved.