Hajela–Smith conjecture for exponential sums of the Liouville function

Let $\bm{\lambda}(n)$ be the Liouville function, defined by $\bm{\lambda}(1)=1$ and $\bm{\lambda}(n)=(-1)^r$ when $r$ is the number of prime factors of $n$ counted with multiplicity. For $N\in\mathbb{N}$ and $|z|=1$, consider the exponential sum

$$L_N(z)=\sum_{k=1}^{N}\bm{\lambda}(k)z^k.$$

Hajela–Smith conjecture. For every $\varepsilon>0$,

$$\|L_N\|_\infty\ll_\varepsilon N^{1/2+\varepsilon}.$$

This is a conjectural square-root-scale bound for Liouville exponential sums. The source notes that Baker and Harman proved the weaker bound $\|L_N\|_\infty\ll_\varepsilon N^{3/4+\varepsilon}$ under the Generalized Riemann Hypothesis, while the conjecture itself is not assigned a resolution in the source.