I'm a layman so please forgive me any misunderstanding or wrong use of terminology.

I understand that theoretically communication using quantum entanglement has to be impossible because it would lead to time travel.

For example suppose you put one particle of an entangled pair on a spaceship making a relativistic roundtrip under constant acceleration. On earth pass 100 years but due to time dilation only 1 year on the spaceship. Arriving back on earth the spaceship would now carry a communication channel 99 years into the past. Therefore, time paradox.

We have tested both entanglement by comparing results of measurements of entangled pairs, as well as time dilation by putting an atomic clock into orbit and found the expected time dilation. Both phenomena are experimentally established to exist.

But have we tested that entangled pairs are entangled across time as is assumed to produce the paradox? Can we put one entangled pair onto a satellite with an atomic clock and confirm that the entangled state change occurs with the expected delay/advance caused by time dilation?