  Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $\mu$
be the M\"obius function and $\mu_2^2(n)$ be the indicator that $n$ is
cubefree. We prove that $f=\mu^2g$ and $f=\mu_2^2g$ have unbounded partial
sums. Our proofs are built upon Klurman and Mangerel's proof of Chudakov's
conjecture, Klurman's work on correlations of multiplicative functions and
Tao's resolution of the Erd\H{o}s discrepancy problem.
