Mossinghoff–Trudgian sign-change conjecture for weighted Liouville sums

Let $\lambda(n)$ be the Liouville function and, for $\alpha\in\mathbb{R}$, define the weighted summatory function

$$L_{\alpha}(x)=\sum_{n\leq x}\frac{\lambda(n)}{n^{\alpha}}.$$

Mossinghoff–Trudgian conjecture. For $0<\alpha<1/2$, the weighted sum $L_{\alpha}(x)$ changes sign infinitely often. The source notes that this has been proved unconditionally for $\alpha=0$ and $\alpha=1$, while no such sign changes had been found computationally in the stated open range.