Addendum to “Unidirectionally invisible potentials as local building blocks of all scattering potentials”

In [Phys. Rev. A 90, 023833 (2014)], we offer a solution to the problem of constructing a scattering potential $v\left(x\right)$ which possesses scattering properties of one's choice at an arbitrarily prescribed wave number. This solution involves expressing $v\left(x\right)$ as the sum of $n\le 6$ finite-range unidirectionally invisible potentials. We improve this result by reducing the upper bound on $n$ from 6 to 4. In particular, we show that we can construct $v\left(x\right)$ as the sum of up to $n=3$ finite-range unidirectionally invisible potentials, unless if it is required to be bidirectionally reflectionless.