The Elliott-type conjecture on correlations of aperiodic multiplicative functions

Let $P_1,P_2\in \mathbb{Z}[m,n]$ be irreducible binary quadratic forms that are not multiples of each other. A pair of forms is good for vanishing of correlations of aperiodic multiplicative functions if the associated correlations vanish in the sense of Definition of the source.

Elliott-type correlation conjecture. If $P_1$ and $P_2$ are irreducible binary quadratic forms that are not multiples of each other, then they are good for vanishing of correlations of aperiodic multiplicative functions.

This conjecture is used for a conditional treatment of partition regularity in cases not covered by the paper's unconditional theorem. It is stated as open even for the Liouville function, for example for the correlation of the values at $m^2+n^2$ and $m^2+2n^2$.