  We discuss a residual freedom of the next-to-leading BFKL eigenvalue that
originates from ambiguity in redistributing the next-to-leading~(NLO)
corrections between the adjoint BFKL eigenvalue and eigenfunctions in planar
$\mathcal{N}=4$ super-Yang-Mills~(SYM) Theory. In terms of the remainder
function of the Bern-Dixon-Smirnov~(BDS) amplitude this freedom is translated
to reshuffling correction between the eigenvalue and the impact factors in the
multi-Regge kinematics~(MRK) in the next-to-leading logarithm
approximation~(NLA). We show that the modified NLO BFKL eigenvalue suggested by
the authors can be introduced in the MRK expression for the remainder function
by shifting the anomalous dimension in the impact factor in such a way that the
two and three loop remainder function is left unchanged to the NLA accuracy.
