Tao's asymptotic Elliott conjecture

Let $\mathbb{D}$ be the closed unit disc, and define the pretentious distance by

$$\mathbb{D}(g_1,g_2;x):=\left(\sum_{p\le x}\frac{1-\operatorname{Re}(g_1(p)\overline{g_2(p)})}{p}\right)^{1/2}.$$

Let $k\ge1$ and let $h_1,\ldots,h_k\in\mathbb{N}$ be fixed and distinct. For $x\ge A\ge2$, let $f_1,\ldots,f_k:\mathbb{N}\to\mathbb{D}$ be multiplicative, and suppose that for every Dirichlet character $\chi$ of modulus at most $A$,

$$\inf_{|t|\le Ax^{k-1}}\mathbb{D}(f_1,\chi(n)n^{it};x)\ge A.$$

Tao's asymptotic Elliott conjecture. Under these hypotheses,

$$\left|\frac1x\sum_{n\le x}f_1(n+h_1)\cdots f_k(n+h_k)\right|=o_{A\to\infty}(1)+o_{x\to\infty}(1).$$

The paper describes this as a corrected asymptotic version formulated by Tao; the supplied text gives no resolution status.