  We investigate a conjecture used in recent experiments to reconstruct the
complete dynamics of Fano autoionisation processes out of measured spectral
amplitudes [Gruson et al, Science 354 734 (2016); Beaulieu el al, Science 358
1288 (2017); Busto et al, J. Phys. B: At. Mol. Opt. Phys. 51 044002 (2018)].
The validity of the conjecture is established analytically within the formalism
of Fano, and tested numerically on model atoms displaying adjustable
autoionising states. A general condition for which the conjecture is valid,
beyond the Fano case, is then derived.
