  We prove three new characterizations of the generalized inverse Gaussian
(GIG), asymmetric Laplace (AL), shifted exponential (sExp) and shifted
truncated exponential (stExp) distributions in terms of non-trivial
independence preserving transformations, which were conjectured by Croydon and
Sasada in \cite{CS1}. We do this under the assumptions of absolute continuity
and mild regularity conditions on the densities.
  Croydon and Sasada \cite{CS2} use these independence preserving
transformations to analyze statistical mechanical models which display KPZ
behavior. Our characterizations show the integrability of these models only
holds for these four specific distributions in the absolutely continuous
setting.
