I think companies like NetJets and FlexJet will buy them on a very large scale once the plane enters production. The reason is simple: unlike the Concorde, the QSST will meet even the stringent ICAO rules on noise and exhaust emissions, thanks to technologies vastly more modern than the technology used on the Concorde (which is essentially late 1950's knowledge of aerodynamics).
By the way, by limiting the top speed to around Mach 1.6, it drastically lessens the need for expensive heat-resistant structural materials, since the structural heating of plane flying at Mach 1.6 is much lower than the Concorde's Mach 2.1 cruising speed.