Conservative or non-conservative? Frame dependent?

In classical mechanics, forces are frame invariant. Work is not, in general, because trajectories are not frame invariant.

However, definition of conservative forces requires only that in each reference frame the work depends only on the initial and final point. Variation of this value with the change of reference frame does not make the definition of conservative field frame dependent.

Even if the null work of a conservative force for a given closed trajectory in one frame could correspond to a non-zero work for the same motion as observed in a different frame, this does not change the property of being conservative, because in all these cases the original closed trajectory is mapped into an open curve and all the paths having the same initial and final point do correspond to the same value for the work.

No mathematics from my side, but here is what my argument is.

One everyday life conservative force is gravity. Suppose you and your friend are diving side by side in your cars (which can fly). For the time being neglect friction and assume earth to be flat. From the perspective of a observer standing on ground bothering of you have the same velocity vector, $\mathbf v = v_x \hat i$. Now suppose that your friend starts to fly up vertically (with respect to you) and then returns back vertically down to the same point (in your reference frame).

So what do you observe?

You observe that neither of his kinetic or potential energy increase/decrease and hence the gravitational force is conservative.

What does that standing observer observe?

He/she sees that their is no difference in kinetic and potential energy of you and your friend. And hence concludes that gravitational force is conservative. Why so? Because, here, you reach your friend reaches his vertically initial position and gravity acts vertically downwards(hence cannot change the kinetic energy into any other perpendicular directions).

The generalization of this to an arbitrary inertial frame and an arbitrary conservative force requires math (which I can't provide you for the time being).

The electric field of a point charge is conservative in the charge's rest frame. In any other frame, this field transforms into a mixture of electric and magnetic fields, in which the electric field is nonconservative (i.e., its curl is nonzero).