Chowla's conjecture on Möbius correlations

Let $r\geq 1$, let $n_1,\ldots,n_r\in\mathbb{N}$, and let $k_1,\ldots,k_r\in\{1,2\}$, with the $k_i$ not all even. Then

$$\sum_{n=1}^N\mu^{k_1}(n+n_1)\mu^{k_2}(n+n_2)\cdots\mu^{k_r}(n+n_r)=o(N)$$

as $N\to\infty$. Chowla's conjecture. The displayed asymptotic holds for every such choice of shifts and exponents. This is a higher-order correlation conjecture for the Möbius function; the source states that its only proven instance is the prime-number-theorem estimate given earlier, so the general statement remains open.