In a previous Phys.SE question, Does a spaceship travelling at near lightspeed see the universe aging slow or fast?, the answer (which was followed by a proof involving co-moving reference frames) was given as
The short answer is that yes, an astronaut moving relative to the cosmic microwave background would measure a shorter time since the Big Bang than an observer stationary wrt to the CMB.
However, an observer in such a spaceship will consider the time of any object which is at the CMBR co-moving reference frame to be moving slower than itself. Is this not a conflicting result?
For example, let's say the spaceship and a CMBR Earth communicate as they pass by each other. Each would have an estimate of the age of the universe, and each would have an estimate of the measured age of universe that the other would give, based on their own measurement of the age and the time dilation that they assume the other would experience. Here are the results
The CMBR observer is fine - both his estimate of the universe's age that the spaceship would give and the estimate actually given by the spaceship match and are less than his own estimate of the universe's age. However, the spaceship expects the CMBR observer to have a lower estimate of the age of the universe because their clock is (according to spaceship observer) ticking slower than his own. What the spaceship observer does not expect is that the CMBR observer's estimate is larger than his own estimate of the age of the universe, yet that is what happens. How is this resolved without implying a preferred reference frame?