Conjecture: If $k \in \mathbb{N}$ and $S$ is an infinite set of primes, then the multiplicative $\pm$-sequence generated by $S$ contains $+^k$ as a substring infinitely often. (If $S$ is allowed to be finite then this fails for example with $S=\{2\}$ and $k=4$.) In other words, we have a totally multiplicative function $f$ such that $f(p) = -1$ whenever $p \in S$ and $f(p) = 1$ for all other primes, so all the other values are determined to be $-1$ or $1$, and we wish to show that there are an infinite number of starting points $x_0$ such that $f(x) = 1$ for all $x \in \{x_0, \dots, x_0 + k - 1\}$.
1. This would imply the multiplicative variant of the EDP, but it is a seemingly stronger statement, so I wonder if there are any counterexamples?
I'm particularly interested in the case $S = \mathbb{P}$. For $k=2$, we are trying to show there are an infinite number of consecutive pairs in which both numbers have an even number of prime factors counting multiplicity, or in terms of the Liouville function $\lambda(x) = \lambda(x+1) = 1$. This case is implied by twin primes but that's not very satisfying.