Let $\lambda$ be the Liouville function. One version of Chowla's conjecture says that for each set of distinct natural numbers $h_1 , \dots , h_k$,

$$\sum_{n\leq x} \lambda(n+h_1) \dots \lambda(n+h_k) = o(x)$$

Is this conjectural behavior stable under perturbations?

Say that a function $f: \mathbb N \to \pm 1$ is a relaxed Liouville function with error $\epsilon$ if for each $n$, $f(nm)=\lambda(n)f(m)$ for a density $1-\epsilon$ set of $m$.

Note that this definition is meaningful for $\epsilon = 0$. Then the obvious questions are:

Does there exist a relaxed Liouville function $f$ with error $0$ and a set of numbers $h_1, \dots, h_k$ such that

$$\sum_{n\leq x} f(n+h_1) \dots f(n+h_k) \neq o(x)?$$

and more vaguely:

Does there exist a relaxed Liouville function $f$ with error $\epsilon$ for $\epsilon$ "small", a set of numbers $h_1, \dots, h_k$, and a "large" constant $\delta $ such that:

$$\left|\sum_{n\leq x} f(n+h_1) \dots f(n+h_k)\right| \not \leq ( \delta +o(1) ) x?$$

For instance, can we take $\delta/\epsilon$ to be arbitrarily large with fixed $h_1, \dots, h_k$?

Clearly negative answers to these questions would imply Chowla's conjecture. So I am hoping instead for positive examples.

My motivation is that many elementary approaches that can prove some weak partial results towards Chowla's conjecture are very stable and therefore apply even for relaxed Liouville functions. I want to understand the limits of these elementary methods by explicit counterexample.

Here is an alternate definition which I think captures the same spirit. Can we find, for some $h_1, \dots, h_k$, for each $N$ a function $f: \mathbb N \to \pm 1$ with $f(pn) = - f(n)$ for $p< N$ such that

$$\left|\sum_{n\leq x} f(n+h_1) \dots f(n+h_k)\right| \not \leq ( \delta +o(1) ) x$$

for some $\delta$ independent of $N$? Or even just some $\delta$ with $\delta \log N$ arbitrarily large?

The idea being that about $1/\log N$ numbers have no prime factors less than $N$ and hence are arbitrary.