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Neutron diffraction and calorimetric measurements are used to study the field-induced quantum phase transition in piperazinium-Cu$_2$(Cl$_{1-x}$Br$_x$)$_6$ ($x=0$, x=3.5% and x=7.5%), a prototypical quantum antiferromagnet with random bonds. The critical indexes $\phi$ and $\beta$ are determined. The findings contradict some original predictions for Bose Glass based on the assumption $z=d$, but are consistent with recent theoretical results implying $z<d$. Inelastic neutron experiments reveal that disorder has a profound effect on the lowest-energy magnetic gap excitation in the system.