The article first studies the propagation of well prepared high frequency waves with small amplitude ε near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, and Tadmor [J. Amer. Math. Soc., 7 (1994), pp. 169-192] about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a definition of smooth nonlinear flux is stated and compared to classical definitions. Then it is proved that the uniform smoothness expected by these authors in Sobolev spaces cannot be exceeded for all smooth nonlinear fluxes.1. Introduction. This paper deals with supercritical geometric optics to highlight the maximal regularizing effect for nonlinear multidimensional scalar conservation laws. This effect is studied in Sobolev spaces by Lions, Perthame, and Tadmor in [23]. They obtain a uniform fractional Sobolev bounds for any ball of L ∞ initial data under a nonlinearity condition on the flux quantified by a parameter α ∈ ]0, 1]. They also conjectured the better Sobolev exponent s = α for entropy solutions.For the first time, the multidimensional case is investigated to bound this maximal smoothing effect. Furthermore, all smooth nonlinear fluxes are considered in this paper. We build sequences of solutions uniformly bounded in W s,1 loc with the conjectured maximal exponent s = α and with no possible improvement of the Sobolev exponent.High frequency periodic solutions are used for this purpose. Near a constant state and for L ∞ data, geometric optics expansions with various frequencies and various phases are validated in the framework of weak entropy solutions and of L 1