Refined conjecture for the half-weighted Liouville sum

Let $\lambda(n)$ be the Liouville function and define

$$L_{1/2}(x)=\sum_{n\leq x}\frac{\lambda(n)}{\sqrt n}.$$

Refined half-weight conjecture. As $x$ tends to infinity,

$$L_{1/2}(x)\sim\frac{\log x}{2\zeta(1/2)}.$$

This refines the preceding conjecture about eventual nonpositivity of $L_{1/2}(x)$ and is motivated in the source by a heuristic for the order of the fluctuation around the displayed main term.