The binary corrected Elliott conjecture for ordinary averages

For multiplicative $f_1,f_2:\mathbb{N}\to\{|z|\le1\}$, at least one satisfying $\inf_{|t|\le N}D(f_j,\chi n^{it};N)\to\infty$ for every fixed Dirichlet character, and fixed distinct $h_1,h_2\ge0$: $\frac1N\sum_{n\le N}f_1(n+h_1)f_2(n+h_2)\to0$. Also for fixed nonproportional affine forms and in fixed residue classes. Functions may be complex and need not be completely multiplicative. Qualitative only, no rate; the $k$-point Elliott conjecture for $k\ge3$ is not addressed.

For multiplicative $f_1,f_2:\mathbb{N}\to\{|z|\le1\}$, at least one satisfying $\inf_{|t|\le N}D(f_j,\chi n^{it};N)\to\infty$ for every fixed Dirichlet character, and fixed distinct $h_1,h_2\ge0$: $\frac1N\sum_{n\le N}f_1(n+h_1)f_2(n+h_2)\to0$. Also for fixed nonproportional affine forms and in fixed residue classes. Functions may be complex and need not be completely multiplicative. Qualitative only, no rate; the $k$-point Elliott conjecture for $k\ge3$ is not addressed.