The half-power sign conjecture for Liouville sums

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function and define

$$L(X,\alpha)=\sum_{n\leq X}\frac{\lambda(n)}{n^\alpha}.$$

Half-power sign conjecture.

$$L\left(X,\frac12\right)=\sum_{1\leq n\leq X}\frac{\lambda(n)}{n^{1/2}}\leq 0$$

for all $X\geq 17$. The inequality has been checked computationally for $17\leq X\leq 300{,}001$, but no proof for all $X\geq 17$ is given.