At the Paul Scherrer Institut (PSI) fission track detectors are routinely used in moderators for determining the ambient dose equivalent in stray radiation fields due to neutrons. At the high-energy field CERF (CERN EU high-energy reference field) at CERN, the response of fission track detectors within two different moderators, where one moderator has an inlay of Pb to increase the response of the detector to high-energy neutrons, was surprisingly similar. To confirm this unexpected behaviour, a measurement campaign in another high-energy field, i.e. CAVE A at the Helmholtzzentrum für Schwerionenforschung (GSI) was carried out. CAVE A at GSI is one of few facilities where a stray radiation field with neutrons above 20 MeV can be generated under controlled conditions. Reference doses were obtained by simulating the stray radiation field in the measurement positions. These simulations were carried out independently by GSI and PSI using two different Monte Carlo codes, i.e. Fluka and MCNPX, respectively. In this paper the measurements at GSI are described in detail. The results are then compared to previous measurement campaigns in the radiation field of CERF as well as ²⁴¹Am-Be and ²⁵²Cf radionuclide sources, which all feature different neutron spectra. None of these campaigns confirm the improvement in sensitivity of the Pb enhanced moderator compared to the PSI standard moderator in combination with fission track detectors.