  The four-dimensional Einstein-Gauss-Bonnet (4DEGB) gravity is proposed as a
singular limit of the higher-dimensional EGB gravity with a rescaled coupling
constant. Follow-up work indicates that the 4DEGB gravity may be equivalent to
a special Horndeski gravity. However, the full equivalence of the two theories
has not been proven. In this paper, we test the possible equivalence for
Bianchi type I metric. We consider two symmetry choices of the extra dimensions
in the 4DEGB gravity and obtain the explicit cosmological evolution equations
of the theories. Our result shows that, for one symmetry choice, there is a
correspondence between the 4DEGB gravity and the special Horndeski gravity.
However, for the other symmetry choice, there is no correspondence between the
two theories. Therefore, the 4DEGB gravity is not equivalent to the special
Horndeski gravity for the general case.
