Now there's a sequel out, which adds a MiG-31 into the inventory. As far as I can tell, the system uses the same solid rocket boosters as does TSR.2, although it launches horizontally from a runway.
By an amazing coincidence, there's a project running right now, between the government of Kazakhstan and Moscow Institute of Thermal Systems (MIT or MITS), to launch a satellite booster from MiG-31. The project name is "Ishim". USSR developed a system in answer to U.S. ASAT, with MiG-31 instead of F-15. After the collapse of USSR, the carrier aircraft were stored at their airfield at the territory of Kazakhstan. So, someone wants to put them to good use now.
The airplane was adorned with additional wingtip fins, because the huge missile created directional stability problems at high Mach numbers. This is something creators of Stratos 4 did not account for, presumably because they did not know that MiG-31D existed.
Tangentially, I think that the project is doomed to failure. Although it's possible that, with Kazakh money, MIT/MITS may create a working booster, its performance is going to be marginal because you cannot hang all that much off the belly of MiG-31. And it's going to be too expensive to compete in international launch market.
Orbital Sciences launches Pegasus from L-1011, and they can only do it because L-1011 is really big. Even then, they put something like 950kg (IIRC) into LEO. It will be a miracle if Ishim does 300kg.
As for the price, this is what Elon Musk said about Pegasus:
There is a fundamental difference in architecture between ourselves and the Pegasus. I think if you were the smartest person on earth you could not make the Pegasus cheap.I happen to think that everything Mr. Musk said is applicable to Ishim, only with additional handicap of a less capable carrier aircraft.
The reason I say that is because it is a five stage rocket. You've got an airplane, which is a dedicated Lockheed L-1011. No matter how many times you launch, you have to maintain that plane at several million dollars per year. You have to have dedicated pilots.
Your range safety is much more complex because you essentially have a man-rated system - you are interacting a rocket and a plane with people on board, and then launching it with maybe 20 or 30 feet separating the pilot and that rocket. So I think that complicates things.
Then you have three solid rocket motor stages, including a complete hypersonic airplane in the first solid rocket stage. And then you have the fifth stage, which is the liquid apogee HAPS stage. So if you were the smartest person on Earth, I don't think you could make that system very cheap.
If you look at ours in contrast: it is a two stage rocket, no wings, no control surfaces, both stages are the cheapest propellant you can use, LOX/Kerosene.
It seems clear that a THAAD-style rocket system would do a much better job than what equipment creators of Stratos 4 envisioned, for much less money. But then, it would not be piloted by teenage girls, and we can't have that!