Two objects forever attracting gravitationally without getting closer, what happens?

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Hello I was wondering what exactly would happen in the following scenario.

Imagine an empty universe except for two equal masses and some mechanism for inflation (dark energy?). The two masses attract each other gravitationally, converting potential gravitational energy into kinetic energy. But what would happen if the inflation of space always remains equal to the speed at which these masses would approach, causing the masses to keep the exact same distance forever.

Does the kinetic energy of both objects forever increase, causing them to approach the speed of light closer and closer eventually?
If this happens, is there anything by which observers on the objects could actually see this happening?
What about an observer viewing from a third mass? (ignore the gravitational effects of the third object on the other ones, and assume that the inflation does not cause the third object to remain stationary compared to the others) Would the increase in kinetic energy cause a stronger bending of spacetime, possibly influencing that third object? If so, can anything be said of the rate at which this influence changes? Is it exponential because more kinetic energy means more gravitation which means more kinetic energy added?
 

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