This is about the observation of the REAL FACTS after the experiment. When they compare their clocks.From which frame? In the frame of the ship, no. In the frame of the earth, yes. There is no absolute standard for time, so this answer depends on the frame n which the comparison is being done.
Stop confusing yourself with the illusion of time dilation which can never be observed and verified. A good example of this illusion of yours is the twin paradox. The stay-at-home twin will predict that the travelling twin ages more. Alas, his prediction is falsified.
Time dilation of proper times can only be observed by two observers by sharing the same rest frame twice and one/both having accelerations between.Time dilation is an effect *between* two reference frames. This can be observed by comparing the results in two reference frames. This is done all the time when looking at decays of atomic particles, but it can also be measured using atomic clocks at ordinary speeds.
THERE IS NO OTHER WAY TO OBSERVE TIME DILATION OF OBSERVED PROPER TIMES.
Of course there is and you must have, otherwise you can make no predictions at all. Each object/observer has their own rate of time. Two observers can compare their times by joining into one single rest frame. Joining into two separate rest frames they can compare their average rates of time that occurred between those two frames when they were in separate rest frames.How many times do I have to point out to you that there is no well-defined *rate of time*?
How many times do I have to tell you that AN OBSERVER IN ONE FRAME CAN NOT MEASURE TIME FOR AN OBSERVER IN ANOTHER FRAME. They can predict their times and this is where the relativistic model fails miserably. The observers end up making contradicting predictions of times for each other and when the facts are observed one or both predictions are falsified.Please give the context within a single post. If the shuttle is moving with respect to the ship, the ship will measure time on the shuttle as running slower. If the shuttle is moving with respect to the earth, then the earth will measure time on the shuttle as running slower. But, the shuttle will also measure times on both the ship and the earth as running slower.
Another thing you seem to miss is the difference between time dilation in fly-bys and proper time in round trips. These are different, although connected concepts.
I'll write a separate post about the experiment...