Reciprocity and self-tuning relations without wrapping

We consider scalar Wilson operators of ${\cal N}=4$ SYM at high spin, $s$, and generic twist in the multi-color 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 $\frac{1}{s(\ln s)^n}$ (inclusive) at any fixed 't Hooft coupling $\lambda$. 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 non-linear integral equation and anomalous dimension expansions at the next order $\frac{(\ln s)^{2}}{s^2}$, at fixed 't Hooft coupling, in such a way to re-establish the aforementioned relation (which fails otherwise).