  We present a systematic method for the derivation of a relation which
connects the correlation function of operators on the straight Maldacena-Wilson
line with the integrability data for the cusp anomalous dimension. As we show,
the derivation requires very careful treatment of the UV divergences. Our
method opens a way to derive infinitely many constraints on integrals of
multi-point correlation functions, relating them with the integrability data
for the generalised cusp anomalous dimension governed by the Quantum Spectral
Curve. Such constraints have been shown recently to be very powerful in
combination with the numerical conformal bootstrap, leading to very narrow
non-perturbative bounds on conformal data beyond the spectrum.
