Which of the following quantities must remain constant for a...

According to Newton's second law, the rate of change of linear momentum is equal to the net external force applied to the system, $\vec{F}_{ext} = \dfrac{d\vec{p}}{dt}$. Since no external force acts on the body, $\vec{F}_{ext} = 0$, which implies $\vec{p} = \text{constant}$. Thus, linear momentum must remain constant.

The absence of any external force also implies that the net external torque acting on the system about any point is zero, $\vec{\tau}_{ext} = 0$. Since $\vec{\tau}_{ext} = \dfrac{d\vec{L}}{dt}$, we get $\vec{L} = \text{constant}$. Thus, angular momentum must also remain constant.

For a nonrigid body, the relative positions of its constituent particles can change. Internal forces can do work (e.g., during expansion or contraction), which leads to a change in the kinetic energy of the system. Therefore, kinetic energy is not necessarily constant.

Similarly, a change in the relative positions of the particles alters the mass distribution of the body, which changes its moment of inertia. Therefore, the moment of inertia is not necessarily constant.