Vanishing-correlation conjecture for distinct irreducible binary quadratic forms

Let $P_1,P_2\in\mathbb{Z}[m,n]$ be irreducible binary quadratic forms that are not multiples of one another. Say that $P_1,P_2$ are good for vanishing of correlations of aperiodic multiplicative functions if, whenever $f_1,f_2\colon\mathbb{N}\to\mathbb{S}^1$ are completely multiplicative and either $f_1$ is aperiodic on the primes associated with $P_1$ or $f_2$ is aperiodic on the primes associated with $P_2$, the correlation averages specified in the paper vanish for every $Q\in\mathbb{N}$ and $a,b\in\mathbb{Z}_+$.

Vanishing-correlation conjecture. If $P_1,P_2\in\mathbb{Z}[m,n]$ are irreducible binary quadratic forms that are not multiples of each other, then they are good for vanishing of correlations of aperiodic multiplicative functions.

The conjecture is intended to supply the aperiodic case needed for an application to partition-regularity results. The paper explicitly places it among its open questions and gives no proof in the supplied text.