Polylogarithmic growth conjecture for the auxiliary function Psi

Define

$$\Psi(x)=\sum_{n\ge1}\frac{2}{\sqrt n}\cos\!\left(\pi\left(\frac{3n-1}{4}-\frac{x}{n}\right)\right).$$

Polylogarithmic growth conjecture. There exists $k>0$ such that

$$\Psi(x)=O\!\left(\log^k|x|\right),\qquad x\to\pm\infty.$$

This is proposed as a stronger property than the preceding interior-region decay conjecture; the limited numerical evidence is compatible with $k=2+\varepsilon$.