Does asymmetric rigid body experience torque-free precession?

Before I address your question specifically:
You are contrasting two things, but I think that supposed contrast isn't actually there.

In the general case of a spinning top that has recieved a sudden jolt:
The resulting motion from that jolt will be a combination of nutation and gyroscopic precession.

The nutation component of the motion is torque-free precession; they're the same thing.

So for instance: it's not that a gyroscope/top will either move in torque-free precession or gyroscopic precession; it can move in a superposition of the two. Whether you do get that combination depends on how you start the motion.

In the case of gyroscopes and spinning tops we rarely see nutation because the usual way of releasing a gyroscope/top is to release it gingerly (as opposed to a sudden jolt). The nutation is a high frequency motion; careful release leaves no room for imparting nutation.

So let me generalize your question to cover both cases:

• Can a cuboid move in torque-free precession?
• Can a cuboid move in gyroscopic precession?

I used the word 'cuboid' here to refer to an object with three different moments of inertia. For instance, a hardcover book (with an elastic band around it so the book won't flip open) is a common object for this kind of demonstration. (It has good size and it has a good weight to it.)

Torgue-free precession
If the object is fairly close to having an axial symmetry the deviation from moving just like an axially symmetric body will be fairly small. The bigger the deviation from axial symmetry, the more the irregularity affects the motion of the object.

So, for instance, I can see a cuboid being thrown up in such a way that slow motion footage will show it to be moving along torque-free precession, just as is seen in footage of a case of say, Feynmans wobbling plate. But I also expect that when the amplitude of the torque-free precession is large you get visible irregularities.

Gyroscopic precession
Generally the angular velocity of gyroscopic precession is slow. (The faster the spin rate the slower the corresponding precession rate.) The practical reason to make a gyroscope wheel circular is to minimize air friction. If air friction is not a concern then roundness isn't important, what remains is that the gyroscope wheel must be well balanced. So, for a cuboid: when spinning around it most stable axis I expect gyroscopic precession to come out very much the same as with an axially symmetric object.