The optimal scale in Linnik's microsquare problem for almost all integers

We prove that, for any positive function $Ψ(t)\to\infty$, almost all positive integers $n$ satisfying $4\nmid n$ and $n\not\equiv7\pmod8$ can be written as $n=x^2+y^2+z^2$, where $x,y,z\in\mathbb{Z}$ and $1\leqslant z<\sqrt{\log n}\,Ψ(n)$. This improves Wooley's bound, and we show that the result is