Relativity 09.06. Four-momentum is Conserved in All Frames

Physics II: Special Relativity. College of the Atlantic. David P. Feldman.
Licensed under Creative Commons: Attribution-NonCommercial-ShareAlike 4.0 International.

[EvelynBonilla-tr]

I don't expect an answer since it has been a couple years now since this video was put up. My question is that 4-momentum is by definition dR/dtau where tau is the proper time of the object in question, however, here there are four different objects, object 1, 2 (before collision) and 3 and 4 (after collision), and you said "rest frame" ... but the "rest frame" of which of the 4 objects? Each of the 4 objects have (potentially) different times since they could be moving at different relative velocities. I'll be grateful if you could answer my question.

[ethannguyen]

Since the t component is kinetic energy, wouldn’t that mean in cases where kinetic energy isn’t conserved, momentum isn’t conserved?