In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case we detail the proof of this conjecture in the framework of Sobolev fractional spaces $W^{ s,1}$ , and in fractional $BV$ spaces: $BV^s$. The $BV^s$ smoothing effect is more precise and optimal. It implies the optimal Sobolev smoothing effect in $W^{ s,1}$ and also in $W^{ s,p}$ with the optimal $ p = 1/s$. Moreover, the proof expounded does not use the Lax-Oleinik formula but a generalized one-sided Oleinik condition.