The logarithmically averaged Chowla conjecture

Let $k\geq 1$, let $a_1,\dots,a_k$ be natural numbers, and let $b_1,\dots,b_k$ be distinct nonnegative integers satisfying

$$a_i b_j-a_j b_i\neq 0$$

for $1\leq i<j\leq k$. Let $\lambda$ denote the Liouville function. Logarithmically averaged Chowla conjecture. For all $2\leq\omega\leq X$,

$$\sum_{X/\omega\leq n\leq X}\frac{\lambda(a_1n+b_1)\dots\lambda(a_kn+b_k)}{n}=o_{\omega\to\infty}(\log\omega).$$

The conjecture is a logarithmically averaged weakening of Chowla's conjecture; its $k=2$ case is known, while the general cases remain open.