Modified Elliott conjecture for non-pretentious multiplicative functions

Let $\mathbb{D}$ be the closed unit disc, let $\mathbb{D}(g_1,g_2;x)$ denote the pretentious distance, and let $\delta^+_{\log}$ denote upper logarithmic density. Let $k\ge1$, and let $f_1,\ldots,f_k:\mathbb{N}\to\mathbb{D}$ be multiplicative functions with $f_1$ non-pretentious. Modified Elliott conjecture. There exists a set $\mathcal{X}\subset\mathbb{N}$, depending only on $f_1$, with $\delta^+_{\log}(\mathcal{X})=1$, such that for every choice of distinct $h_1,\ldots,h_k\in\mathbb{N}$, the correlation average tends to zero along $x\to\infty$ with $x\in\mathcal{X}$:

$$\lim_{\substack{x\to\infty\x\in\mathcal{X}}}\frac1x\sum_{n\le x}f_1(n+h_1)\cdots f_k(n+h_k)=0.$$

This is proposed as an asymptotic replacement for Elliott's original conjecture, whose unrestricted formulation is stated in the paper to be false.