Generalized Sun conjecture for the summatory function of $(-a)^{\Omega(n)}$

For real $a>0$, let $\Omega(n)$ denote the number of prime factors of $n$, counting multiplicity, and define

$$W_a(x)=\sum_{n\leq x}(-a)^{\Omega(n)}.$$

For $a\geq 2$, the function has jumps of size $x^{\log a/\log 2}$ at powers of $2$, so for infinitely many $x$,

$$|W_a(x)|\geq\frac{x^{\log_2 a}}{2},$$

where $\log_2 x=\frac{\log x}{\log 2}$. Generalized Sun conjecture. For all real numbers $a\geq 2$ and sufficiently large $x\geq x_0(a)$,

$$|W_a(x)|<x^{\log_2 a}.$$

This generalizes Sun's conjecture from $a=2$ and is motivated by computations suggesting that the normalized function $W_a(x)/x^{\log_2 a}$ remains in $(-1,1)$ for sufficiently large $x$. The conjecture remains open.