  Recently, Arman, Bondarenko, and Prymak constructed a constant width body in
$\mathbb{R}^n$ whose illumination number is exponential in $n$. In this note,
we improve their bound by generalizing the construction. In particular, we
construct a constant width body in $\mathbb{R}^n$ whose illumination number is
at least $(\tau+o(1))^n$, where $\tau\approx 1.047$.
