Will two entangled qubits be affected by gravity?

Let's start with the Bell state $frac1{sqrt2}(|01rangle+|10rangle)$. Let's further assume that the entire universe (or at least the parts of relevance) are pervaded by a static magnetic field along the $z$-axis, which I relate to the unitary evolution $exp(-itH_z)$.

Now let's place one part of the entangled state next to a massive object, such that its own time evolves slower than ours far away.

As you can see in this QUIRK circuit here, the Bell states changes back and forth between $$frac1{sqrt2}(|01rangle+|10rangle)leftrightarrowfrac{e^{iphi}}{sqrt2}(|01rangle-|10rangle), $$ as time goes by with two different speeds. Now you have to come up with a measurement protocol to distinguish these two states by local operations and communications (don't enter the event horizon!)

Whereas you only catch up a global phase which is not measurable...

UPDATE

There is "Space QUEST (QUantum Entanglement Space Test) mission proposal" at the arxiv, to "...Experimentally test decoherence due to gravity". From the abstract:

Some speculative theories suggest that quantum properties, such as entanglement, may exhibit entirely different behavior to purely classical systems.