Both of your true/false statements are true. As the stone slides down the hill, its momentum increases, so its momentum is not conserved.
Deciding on whether momentum is conserved or not depends critically on what your system is. If there are no forces on the system from outside the system, then momentum is conserved. If there is a net force acting on the system from the outside, then the momentum will change with time.
So for the stone going down the hill, the wording of the question means that we want to consider the stone by itself as the system. There is a net outside force (the force from the earth), so momentum is not conserved for the stone.
Momentum is conserved if we had chosen (stone+earth) as a system. The change in the stone's momentum is equal in magnitude and opposite in direction to the change in the earth's momentum. (The earth pulls on the stone and the stone pulls on the earth, in addition to the action/reaction normal forces between stone and hill.) The momentum of (stone+earth) is conserved as the stone slides down the hill.
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