Increase moment of inertia, slipping down an incline

Hi, I'm trying to find out what effect an increase in moment of inertia of a cylinder would have on slipping when the cylinder rolls down an incline.
Starting with: ma=mgsin(theta)-f, where f is the friction force, which is less than or equal to µmgcos(theta), normal force times µ.
We already know that an increase in moment of inertia without increasing the radius will decrease the linear acceleration, and therefore, for instance a hoop, will roll down the incline slower than a cylinder of equal mass and radius.
Therefore ma will decrease, but mgsin(theta) is constant, so f has to increase, right? An increase in moment of inertia requires a higher friction force.
Does this mean that if we were to let a solid cylinder and a hoop race down a ramp of angle (theta), the hoop would start to slip first if we were to increase the angle (theta)? If this is true, wouldn't it be possible for the hoop to win the race, if at a specific angle, the hoop started to slip and the cylinder didn't, because less friction is needed to sustain its rolling motion?
Could anyone please confirm my train of thoughs? Thanks :)