Elliott's conjecture on correlations of multiplicative functions

Let $\mathbb{U}:=\{z\in\mathbb{C}:|z|\leq 1\}$. Let $g_1,\ldots,g_k:\mathbb{N}\to\mathbb{U}$ be multiplicative functions and let $a_1,\ldots,a_k$ be distinct non-negative integers. Assume that there is an index $1\leq j_0\leq k$ such that, for every fixed Dirichlet character $\chi$,

$$\lim_{X\to\infty}\inf_{|t|\leq X}\sum_{p\leq X}\frac{1-\operatorname{Re}(g_{j_0}(p)\overline{\chi}(p)p^{-it})}{p}=\infty.$$

Elliott's conjecture. Then, as $X\to\infty$,

$$\sum_{n\leq X}g_1(n+a_1)\cdots g_k(n+a_k)=o(X).$$

The conjecture is invoked to upgrade the paper's results on chains of Fourier coefficients from upper-density statements to natural-density statements; it is not proved in the source.