Low-dimensional quantum spin systems with introduced disorder are a fascinating playground for investigations of the interplay between quantum effects, strong correlations and disorder. In case of a spin-1/2 chain with random spin interactions, low-temperature theory predicts a ground state of a random-singlet state (RSS). BaCu2(Si0.5Ge0.5)2O7 has been celebrated as an almost ideal realization of the random AF Heisenberg spin chain, however there is to this day no spectroscopical evidence of RSS in this material. Here our Quantum Monte Carlo in connection with magnetization data offers an explanation. They show that the RSS should appear below a cutoff energy of ~0.3-0.5 meV. Since this energy region has been completely ignored in previous studies, we hereby propose to study the low-energy scattering coming from singlet-triplet transitions of spatially frozen dimers residing in the RSS.