We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type˜ C2 for all choices of positive weight function. Our approach to computing Lusztig's a-function is based on the notion of a " balanced system of cell representations ". Once this system is established roughly half of the conjectures P1–P15 follow. Next we establish an " asymptotic Plancherel Theorem " for type C2, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig's conjectures for all rank 1 and 2 affine Weyl groups for all choices of parameters.