Chowla's sign-pattern conjecture for consecutive Liouville values

Let $s\geq 1$ and let $(\epsilon_1,\ldots,\epsilon_s)\in\{-1,1\}^s$ be a sign pattern. Chowla's sign-pattern conjecture. The set

$$\{n\in\mathbb{N}:\lambda(n+i)=\epsilon_i\ \forall i\}$$

has natural density $2^{-s}$. This is the sign-pattern form of Chowla's conjecture; the case $s=1$ is equivalent to the prime number theorem, while the full assertion remains open over the integers.