  We prove that for a large class of multiplicative functions, referred to as
generalized divisor functions, it is possible to find a lower bound for the
corresponding variance in arithmetic progressions. As a main corollary, we
deduce such a result for any $\alpha$-fold divisor function, for any complex
number $\alpha\not\in \{1\}\cup-\mathbb{N}$, even when considering a sequence
of parameters $\alpha$ close in a proper way to $1$. Our work builds on that of
Harper and Soundararajan, who handled the particular case of $k$-fold divisor
functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$.
