The free end of a simple pendulum is attached to the ceiling of a...

When the box falls freely, its acceleration is $g$ downwards.

In the frame of reference of the box, a pseudo force $mg$ acts upwards on the bob, which exactly cancels the true weight $mg$ acting downwards.

Thus, the effective acceleration due to gravity in the frame of the box is $g_{\text{eff}} = 0$.

At the lowest point of oscillation, the bob has a horizontal velocity $v$ relative to the box.

Since $g_{\text{eff}} = 0$, there is no tangential force to change the speed of the bob. The tension in the string provides the necessary centripetal force $T = \dfrac{mv^2}{L}$.

With a constant speed and a fixed distance from the point of suspension, the bob will move in a circular path.

Answer: will go in a circular path