We study post-inflationary evolution in $\alpha$-attractor T-models of
inflation. We consider the dynamics of both scalar fields present in these
models: the inflaton and the spectator, as a negative field-space curvature may
lead to geometrical destabilization of the spectator. We perform
state-of-the-art lattice simulations with a dedicated numerical code optimized
for those models. We corroborate earlier findings that the perturbations of the
spectator field are much more unstable than the perturbations of the inflaton
field, so the dynamics of early stages of preheating is dominated by the
evolution of spectator perturbations. We also calculate the spectrum of
gravitational waves originating from scalar fluctuations in order to determine
if the $\alpha$-attractor T-models can be constrained or even ruled out by
present cosmological observations, but not by direct searches of gravitational
waves.
