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Does the inverse square law have anything to do with conservative behavior of the central forces?

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Not really, no. A field is conservative if it can be written as the gradient of a potential function:

$$\mathbf{F}=\nabla f$$

You could equally well define the potentials $f=kr^2$, or $f=kr^5$, etc., which will produce vector fields that are respectively linear and quartic with $r$.

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All central forces (where central means spherically symmetric, that is independent of angular variables) are conservative: see the quesiton Are all central forces conservative? Wikipedia must be wrong.

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