Linnik's conjecture on small coordinates in representations by three squares

Let $n$ be a positive integer satisfying $n\not\equiv 0,4,7\pmod{8}$. A representation of $n$ by three integer squares is a triple of integers $(x,y,z)$ satisfying

$$x^2+y^2+z^2=n.$$

Linnik's conjecture. For each such $n$, there exists a representation with

$$|z|=O(n^\epsilon)$$

for any $\epsilon>0$.

This is a conjecture on primitive representations of integers by three squares and is connected with small-scale equidistribution of rational points on the sphere. The supplied source gives no resolution status, so it is recorded as open.