Modified wave operators are unbounded on $L^p$ for $p\neq 2$

We study the $L^p$-boundedness of Isozaki--Kitada type modified wave operators associated with Schrödinger operators $P=-\partial_x^2 + V$ in one dimension for potentials $V$ including long-range potentials such as negative Coulomb-like ones $V(x) = -(1+|x|^2)^{-μ/2}$ for $μ>0$. For such potentials,