= Solution
Put $I_1=I_2=I_\perp$, with $I_3\ne I_\perp$. Then
$$
\boxed{p^2=-\Omega^2\frac{(I_3-I_\perp)^2}{I_\perp^2}<0,\qquad
\nu=|\Omega|\frac{|I_3-I_\perp|}{I_\perp}.}
$$
Rotation about the symmetry axis is stable whether that axial moment is larger or smaller than the transverse moment. In fact $\omega_3$ and $\omega_1^2+\omega_2^2$ are exactly constant for a torque-free axisymmetric body: the transverse components rotate into one another at the stated body-frame frequency. Stability here means bounded wobble, not relaxation back to perfect axial rotation.
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