Corradi–Katai conjecture for the binary Liouville Goldbach sum

Prove that, for the binary Liouville Goldbach sum \(\mathcal{G}_2(N)=\sum_{a+b=N}\lambda(a)\lambda(b)\), one has \(\lvert\mathcal{G}_2(N)\rvert=o(N)\) as \(N\to\infty\), equivalently \(\lim_{N\to\infty}\lvert\mathcal{G}_2(N)\rvert/N=0\).

Background

The paper defines $\mathcal{G}_2(N)$ as the sum of $\lambda(a)\lambda(b)$ over positive integers $a,b$ satisfying $a+b=N$, where $\lambda$ is the Liouville function. The Corradi–Katai conjecture asks for cancellation of this binary convolution at the scale of the trivial bound $\lvert\mathcal{G}_2(N)\rvert\leq N-1$.

The paper proves only an averaged result: the set of integers $N$ for which $\lvert\mathcal{G}_2(N)\rvert>\delta N$ has density zero for every fixed $\delta>0$, yielding $$\liminf_{N\to\infty}\lvert\mathcal{G}_2(N)\rvert/N=0$$. It explicitly notes that the full conjecture had resisted an unconditional proof, although it had been proved conditionally on the existence of infinitely many Siegel zeros.