Suppose you had two entangled photons, and dropped one of them down a gravity well. The dropped photon would blueshift, but would the entangled photon of the pair blueshift as well? Would it remain the same frequency?
What you are really asking about is time-energy entangled photons. The expression we use is frequency bin entanglement.
Time-energy entanglement can also be viewed as frequency entanglement, as demonstrated in recent works [17, 18].
First, a laser pump produces frequency entangled photon pairs. The frequency of each photon is uncertain, but the sum of their frequencies is well defined. Second, two photons, whose frequency is so close that it cannot be distinguished, belong to the same frequency bin. Third, different frequency bins are made to interfere using electro optic phase modulators.
Now it is very important to understand that when one of the entangled photon pair in your case is moving in a gravity well, this will not destroy entanglement. The two photons have a common wavefunction that describe them both, and as you can see from frequency bin entanglement, the sum of their frequencies is well defined.
Now you are asking about a gravity well.
Another equivalent explanation is that the energy of the photon is decreased, and since E=hf the reduced energy necessarily implies a reduced frequency. The energy of a photon is reduced as it travels upwards because as the photon goes up it must trade some of its EM energy for gravitational potential energy.
In case of frequency entanglement, as you can see, the sum of the frequencies of the photons is well defined, and upon measurement, since the two photons are entangled, measuring the photon in the gravity well will determine the outcome for both photon's frequency measurement.