And, here's the solution my teacher gave: read from top to bottom (and note that the angle $θ$ is the instantaneous slope at that point) and ignore the green highlighting, I apologize for the poor picture quality.
According to his answer, no matter what the path of the particle is, as long as it is acted upon by an external agent that continuously applies a force tangential to the local curvature of the incline without changing the kinetic energy of the particle, the work done by friction remains constant. In this sense, friction almost behaves like a conservative force since its work only depends on the height and the length of the distance covered by the incline, irrespective of the path taken.
Firstly, I would like to verify whether this solution is correct because when I looked at similar problems, the answer only accounts for the potential energy change of $mgH$ and does not have the extra term $μmgL$ in some places, whereas in other problems, the work done by friction appears to be proportional to the actual distance the particle traverses. Can these answers be correct as well? If not, then how would the question have to be modified to make these answers correct?
My second broader question is based on the conservative behavior of friction in this scenario;
Is there perhaps a category of forces that are "pseudo-conservative" i.e. behave conservatively under special circumstances, and if so, can you give an example? Alternatively, are all non-conservative forces capable of behaving conservatively under the correct set of circumstances? Could you again give an example, perhaps with regard to the magnetic force?
