  We prove Lusztig's conjectures ${\bf P1}$-${\bf P15}$ for the affine Weyl
group of type $\tilde{C}_2$ for all choices of positive weight function. Our
approach to computing Lusztig's $\mathbf{a}$-function is based on the notion of
a `balanced system of cell representations'. Once this system is established
roughly half of the conjectures ${\bf P1}$-${\bf P15}$ follow. Next we
establish an `asymptotic Plancherel Theorem' for type $\tilde{C}_2$, 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.
