Recover signal from its spectral amplitude and envelope
One counterexample is two wave packets of identical spread and amplitude, well separated in time so that their envelopes do not overlap, and of high enough (differing) well-separated frequencies such that their analytic signals can be considered Gaussian-enveloped complex exponentials. The Hilbert envelope will thus consist of two identical well-separated Gaussians. The amplitude spectrum will consist of well-separated Gaussians of identical magnitude. There is not enough information to assign the frequencies to the wave packets.



So in general you cannot resolve the original signal, not even up to an ambiguity of a constant phase shift over all frequencies.

With sufficiently large time spreads of the two wave packets their frequencies need not be far apart to still be well separated in the spectrum, satisfying your constraint for a narrow-banded signal.