$\bm{\lambda}$-$L^p$ conjecture for Liouville exponential sums

Let $\bm{\lambda}(n)$ be the Liouville function, and for $N\in\mathbb{N}$ define

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

For a fixed exponent $p$, let $\|L_N\|_p$ denote its $L^p$ norm on the unit circle. $\bm{\lambda}$-$L^p$ conjecture.

$$\Big\|\sum_{k=1}^{N}\bm{\lambda}(k)z^k\Big\|_p\ll_\varepsilon N^{1/2+\varepsilon}.$$

The source presents this as an analogue of the Hajela–Smith conjecture and immediately proves that a corresponding bound for sufficiently large $p>2$ would imply the Riemann Hypothesis. It does not specify the range of $p$ or resolve the conjecture.