Linnik's microsquare conjecture

Let $\varepsilon>0$. For each sufficiently large odd squarefree integer $n\not\equiv7\pmod{8}$, consider representations of $n$ by three integer squares. Linnik's conjecture. There exists a solution $(x_1,x_2,x_3)\in\mathbb{Z}^3$ of

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

with $|x_3|\leq n^{\varepsilon}$. This is a diophantine conjecture asserting that every sufficiently large admissible odd squarefree integer has a representation as two squares and a microsquare; the source describes the paper as making progress toward it but does not state a resolution.