Order of magnitude of the error term $E(x)$ in the count of squarefree integers (Erdős #969)

Let $Q(x)$ count the squarefree integers in $[1,x]$. It is classical that $$Q(x)=\frac{6}{\pi^2}x+E(x).$$ Determine the order of magnitude of the error term $E(x)=Q(x)-\frac{6}{\pi^2}x$ as $x\to\infty$ — i.e. pin down its true growth rate (for instance, decide whether $E(x)=x^{1/4+o(1)}$).

Acceptance. FULLY RESOLVES: a proof (machine-checkable in Lean/Coq preferred, else a complete written proof) establishing the true order of magnitude of $E(x)$ — matching upper and Ω lower bounds up to the stated tolerance (e.g. $E(x)=x^{1/4+o(1)}$, which would also yield the Riemann Hypothesis), or any rigorously proven order. ADVANCES (each requires proof or a reproducible certificate, and must strictly beat the corresponding bound stated in the background): (a) improve the unconditional upper-bound exponent below the $1/2-o(1)$ of Walfisz; (b) improve the RH-conditional upper-bound exponent below the $11/35$ of Liu; (c) sharpen the Ω-result beyond $E(x)\gg x^{1/4}$; or (d) extend the record extremal values of $|E(x)|/x^{1/4}$ via a reproducible high-precision computation with a certificate of correctness. Deliver the proof/formalization, or the search program plus the attained records.