  We explore the cosmological implications of the local limit of nonlocal
gravity, which is a classical generalization of Einstein's theory of
gravitation within the framework of teleparallelism. An appropriate solution of
this theory is the modified Cartesian flat cosmological model. The main purpose
of this paper is to study linear perturbations about the orthonormal tetrad
frame field adapted to the standard comoving observers in this model. The
observational viability of the perturbed model is examined using all available
data regarding the cosmic microwave background. The implications of the
linearly perturbed modified Cartesian flat model are examined and it is shown
that the model is capable of alleviating the $H_0$ tension.
