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Yes. Eqn (iii) can be written as :
$T_1 - T_2 = \frac{I\alpha}{R^2}$.
If the pulley is not massless ($I\ne0$) then $T_1\ne T_2$. The string does not slip against the rim of the pulley so $\alpha \ne 0$ also (assuming that $M_1 \ne M_2$).