Hardy–Littlewood–Chowla conjecture for prime and Möbius correlations

Let $x\to\fty$. Let $a_0,a_1,\ldots,a_{k-1}$ be a fixed admissible $k$-tuple, and let $b_0<b_1<\cdots<b_{l-1}$ be a fixed subset of small integers. Hardy–Littlewood–Chowla conjecture. One has

$$\sum_{n\leq x}\Lambda(n+a_0)\Lambda(n+a_1)\cdots \Lambda(n+a_{k-1})\mu(n+b_0)\mu(n+b_1)\cdots \mu(n+b_{l-1})=o(x).$$

This conjecture combines prime-tuple correlations from the Hardy–Littlewood conjecture with Möbius correlations from the Chowla conjecture. Its asserted asymptotic remains open in general.