Generalized Chowla's conjecture for weighted Möbius correlations

Let $(h_1,h_2)$ be a fixed admissible $2$-tuple. For a positive integer $n$ and $y\ge2$, let $\omega_+(n,y)$ and $\omega_-(n,y)$ count prime divisors of $n$ that are respectively at least $y$ and less than $y$. On squarefree integers define

$$\tau^{\pm}_{\kappa_1,\kappa_2}(n,y)=\kappa_1^{\omega_-(n,y)}\kappa_2^{\omega_+(n,y)}.$$

Generalized Chowla's conjecture. For fixed $0\le\kappa_1<\kappa_2$ and $y=\exp((\log x)^\delta)$ with fixed $0<\delta<1$,

$$\sum_{N/2\le n\le N}\mu(n+h_1)\mu(n+h_2)\tau^{\pm}_{\kappa_1,\kappa_2}(n+h_1,y)\tau^{\pm}_{\kappa_1,\kappa_2}(n+h_2,y)=o\left(\sum_{N/2\le n\le N}\tau^{\pm}_{\kappa_1,\kappa_2}(n+h_1,y)\tau^{\pm}_{\kappa_1,\kappa_2}(n+h_2,y)\right).$$

This asserts cancellation of two-point Möbius correlations after applying the specified preliminary-sieving weights. The source proposes it as a more general weighted form of Chowla's conjecture and provides no proof or resolution.