We consider scalar Wilson operators of N = 4 SYM at high spin, s, and generic twist in the multicolor limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain “reciprocity” and functional “self-tuning” relations up to all terms 1/s(ln s)n (inclusive) at any fixed ’t Hooft coupling λ. Of course, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order in s. On this basis we give some evidence that wrapping corrections should enter the nonlinear integral equation and anomalous dimension expansions at the next order (ln s)2/s2, at fixed ’t Hooft coupling, in such a way to reestablish the aforementioned relation (which fails otherwise).