A non-uniform solid cylinder of mass \(M\) and radius \(R\) has a rotational inertia about its central symmetry axis given by \(I = \beta M R^2\), where \(\beta\) is a positive dimensionless constant. The cylinder is placed on an incline of angle \(\theta\) above the horizontal and released from rest. The cylinder rolls down the incline without slipping. Which of the following expressions represents the minimum coefficient of static friction \(\mu_s\) between the cylinder and the incline required to prevent slipping?