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Consider two places next to each other: Place 1, where there is a gravitational field whereas Place 2 - there's no field.

Now if we lifted a box in place 1, it gains potential energy. Then, we move this box horizontally to place 2. What happens to this energy?

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Your assumption is very wrong. Gravity is everywhere. Or simply, the box has mass. It exerts its very own gravity. If your 1st location has the field, then the 2nd location is also influenced by it unless it is at $\infty$. You can't just shield anything from gravitational field. So, Energy is still conserved when you're moving the box from one place to another...

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