Generic Newton Slopes for Artin-Schreier-Witt Tower in two variables

We prove that for any generic polynomial $f$ in two variables of degree $(d_1,d_2)$ over the rationals, for $p$ large enough the Newton slopes of the character power series $C_f^{\ast }({\chi }_m,s)$ of $f$ at $p$ is independent of the choice of the character ${\chi }_m$ (of conductor $p^m$); and the Newton slopes of the $L$-function $L_f^{\ast }({\chi }_m,s)$ of $f$ at $p$ is in weighted arithmetic progression.