The logical answer is a combined-cycle engine, but there are slim pickings here. The X-43 used a combined-cycle rocket/scramjet, but had to be dropped from a B-52. It did hit around Mach 9.6 though
The Chimera engine is a turbojet/ramjet engine under active development (it or something similar is likely going to power the NGAD), but it most likely won't go faster than Mach 5-6.
They aren't considering a couple important realities though.
First, the US government isn't letting advanced engines get put into civilian airplanes. There's just too great a risk of them falling into Chinese or Russian hands and giving their fighter programs a massive leap forward.
Second, physics aren't in their favor. A large passenger plane means lots of air resistance and lots of weight. Getting all that moving at Mach 2 in the Concorde was a huge task on its own. Moving up to Mach 9 increases the number of issues exponentially. It takes decades to get this working with a tiny plane (eg, all the NASA experiments). The time to scale that up would be even more.
Third, there is ZERO economic possibilities here. The total SR-71+A-12 development costs have been estimated at around $16.5B dollars not to mention $200k per hour to fly. Let's say this company makes 100 of these planes at a similar cost (despite them being much more exotic and larger than the SR-71). The planes will cost $165M each and probably only hold less than 30 passengers/crew. If they are selling 25 tickets and flying that 1-hour flight at $200k/hr, tickets would cost $8,000 just to cover operating expenses. If they expect 20,000 flight hours per plane, that's an additional $8,250 per ticket. I'd guess minimum ticket price would be about $25,000 using these numbers.
But in reality, there's not a market for 100 of these planes. The R&D costs would probably be much higher and the cost per flight hour would certainly be massively higher too.
It's all snake oil.
Stargazer will take off with conventional jet engines, but then transition to rockets once it reaches altitude.
The guys who designed this are obviously smart, much smarter than me, but it's clear that they've taken some 'creative liberties' to design the concept.
My suspicion is that the real product here is the rotational-detonation-engine, the entire "hypersonic passenger vehicle" is just a flashy sales pitch to drive investment & hype towards the development of a genuinely innovative engine that otherwise wouldn't have this amount of funding poured into it.