Note on illuminating constant width bodies

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 $\tau\approx 1.047$.