Linnik's microsquares conjecture for sums of three squares

Let $\varepsilon$ be a positive number, and suppose that $n$ is odd, $n\not\equiv 7\pmod{8}$ and $n$ is squarefree. Consider representations of $n$ as a sum of three integral squares,

$$n=x_1^2+x_2^2+x_3^2.$$

Linnik's microsquares conjecture. Whenever $n$ is sufficiently large in terms of $\varepsilon$, this equation possesses a solution in which

$$|x_3|\leq n^\varepsilon.$$

The conjecture asks for a representation satisfying a local condition together with an exceptionally small one of the three squares; Linnik proved the related equidistribution result assuming the Generalised Riemann Hypothesis. The precise microsquares assertion is presented here as Linnik's conjecture.