In 1994, Lions, Perthame and Tadmor conjectured an optimal smoothing effect for entropy solutions of multidimensional scalar conservation laws. This effect estimated in fractional Sobolev spaces is linked to the flux nonlinearity. In order to show that the conjectured smoothing effect cannot be exceeded, we use a new definition of a nonlinear smooth flux which proves efficient to build bespoke explicit solutions. First, one-dimensional solutions are studied in fractional BV spaces which turn out to be optimal to encompass the smoothing effect: regularity and traces. Second, the multidimensional case is handled with a monophase solution and the construction is optimal since there is only one choice for the phase to reach the lowest expected regularity.