Think twice
If in a system, all the internal forces are conservative, then the mechanical energy of that system remains constant only when there are no external forces acting on it. Is this statement always true?
(The total mechanical energy of a system, is the sum of its kinetic energy and potential energy.)
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The first thing to keep in mind is that potential energy is defined corresponding to the internal conservative forces of a system. It is so because if potential energy were defined for non-conservative forces then we could associate two different energies of an object at the same point in space. Take for example friction which is a non-conservative force. If it was considered for determining the potential energy of an object then that object initially would have a certain potential energy say E, and after traversing a path and coming back to its original position it would have a reduced potential energy which is less than E since friction did some work on it. But then potential energy would be a useless concept. That is why only internal conservative forces are considered to determine the potential energy of a system.
Keeping that in mind, now the change in potential energy of a system is equal to the work done by the internal conservative forces as it passes through one configuration to another.
U f − U i = − W where the negative sign is a convention.
By the work-energy theorem we know that work done is equal to the change in kinetic energy.
So,
U f − U i = − ( K f − K i ) ⇒ U f − U i = K i − K f ⇒ U f + K f = U i + K i
After close inspection it is clear that either there are no external forces or the work done by them is zero . Either of the two is possible so the statement in the question, in general, is false.