  We consider the highest anomalous dimension operator in the SU(2) sector of
planar ${\cal N}=4$ SYM at all-loop, though neglecting wrapping contributions.
In any case, the latter enter the loop expansion only after a precise
length-depending order. In the thermodynamic limit we write both a linear
integral equation for the Bethe root density and a linear system obeyed by the
commuting charges. Consequently, we determine the leading strong coupling
contribution to the density and from this an approximation to the leading and
sub-leading terms of any charge $Q_r$: it scales as $\lambda ^{1/4-r/2}$, which
generalises the Gubser-Klebanov-Polyakov energy law. In the end, we briefly
extend these considerations to finite lengths and 'excited' operators by using
the idea of a non-linear integral equation.
