To sustain circumnavigating motion a centripetal force is required.
The case of a rigid disk that is rotating around non-principal axis often referred to as Feynman's wobbling plate
In the case of Feynman's wobbling plate momentum is internally relocated continuously. That is, in the case of Feynman's wobbling plate there are significant internal stresses.
If the disk is flexible enough to be flexed by the internal stresses then the kinetic energy associated with the wobbling motion dissipates, and as a result the axis of rotation will settle on the principal axis.
Going back to the statement at the start of this answer:
To sustain circumnavigating motion a centripetal force is required.
In the case of a spinning disk the central force that is available is the structural integrity of the disk, which acts in the plane of the disk. The process of dissipating the kinetic energy associated with the wobbling motion is a process of bringing the rotation into alignment with the available centripetal force. Once that state of alignment is reached there is no more opportunity to dissipate kinetic energy.