Distance of a source vs its isotropic HEN energy plan showing the probable 90% interval for the source distance, for different populations and spectral indices. Top: after the first notice, when the number of observed HEN event is 1, for a CCSN/SFR population assuming a spectral index γ = 2, in green. The yellow and blue region shows the region excluded after the first notice, given the minimum and maximum upper limits on the average number of HEN events. Bottom: Same quantities after the second notice reporting the search of additional HEN events and fluence limits in two different time windows (1000s and 2 days). In this plot, the limits are the ones given for the event ICECUBE-230306A. The colored vertical bands show typical maximum energies for different classes of possible sources. The green dashed line indicate the median distances. The magenta lines show the modified 5%–95% interval for a GRB-population together with γ = 2.5, whereas the black lines refer to a population with no redshift evolution and γ = 2.