  The layered van der Waals material ZrTe$_5$ is known as a candidate
topological insulator (TI), however its topological phase and the relation with
other properties such as an apparent Dirac semimetallic state is still a
subject of debate. We employ a semiclassical multicarrier transport (MCT) model
to analyze the magnetotransport of ZrTe$_5$ nanodevices at hydrostatic
pressures up to 2 GPa. The temperature dependence of the MCT results between 10
and 300 K is assessed in the context of thermal activation, and we obtain the
positions of conduction and valence band edges in the vicinity of the chemical
potential. We find evidence of the closing and subsequent re-opening of the
band gap with increasing pressure, which is consistent with a phase transition
from weak to strong TI. This matches expectations from ab initio band structure
calculations, as well as previous observations that CVT-grown ZrTe$_5$ is in a
weak TI phase in ambient conditions.
