The gauge coupling unification in Grand Unified Theories based on the group $E_8$

We consider a theory with the gauge group $E_8$ assuming that the gauge symmetry breaking pattern is $E_8 \to E_7 \times U_1 \to E_6 \times U_1 \to SO_{10} \times U_1 \to SU_5 \times U_1 \to SU_3 \times SU_2 \times U_1$ and vacuum expectation values are acquired only by components of the representations 248. It is demonstrated that in this case there are several options for the relations between the gauge couplings of the resulting theory, but only one of them gives $\alpha_3 = \alpha_2$ and $\sin^2\theta_W = 3/8$. Also, it is the only option for which the resulting theory can include all MSSM superfields.