I may sound a bit presumptuous to you, but why worry about the implications of GR being wrong when it has been experimentally and theoretically confirmed over and over?

I will attempt to show how the altitude thing embraces GR:

(I am still a student and am not as well-versed in physics as alexander or bruce are, so anybody feel free to correct any errors in my logic)

Let's say you have a circle that spins about its center. A point at a distance r from the center of a circle would have a higher angular velocity than a point at distance r/2 from the center of a circle. If this were not true, the circle would look more like a whirlpool or vortex. The relationship here is shown by the equation a=(v^2)/r, or r = (v^2)/a.

As something gets closer to the axis of rotation, the angular veocity increases (a similar topic is the classic example where an ice skater spins and alternates spreading her arms out and bringing them in).

As the equation shows, as r increases, v increases. Another dimension can effortlessly be added to this by making the circle a sphere. Now we have the earth, pretty close to a sphere, rotating about its axis. Using the equation r = (v^2)/a, we can easily see that the farther you are away from the center of the earth, the higher your angular velocity. Of course, this would be easiest to calculate along the equator, but the calculations can be done anywhere on earth (they would just be considerably more time-consuming).

Relativity states that the faster an object goes, the slower time passes for it (reference frames notwithstanding). An object is moving faster farther away from the center of the earth so time passes slower for it.