= Solution
First consider rotation predominantly about the third <principal axis>, writing $\omega_3=\Omega+O(\omega_1^2+\omega_2^2)$. To first order the third Euler equation gives $\dot\Omega=0$, while the transverse components satisfy
$$
\dot\omega_1=\frac{I_2-I_3}{I_1}\Omega\omega_2,
\qquad
\dot\omega_2=\frac{I_3-I_1}{I_2}\Omega\omega_1.
$$
Therefore
$$
\ddot\omega_1
=-\Omega^2\frac{(I_3-I_2)(I_3-I_1)}{I_1I_2}\omega_1,
$$
and the same equation holds for $\omega_2$.
Order the principal moments as $I_1<I_2<I_3$. The coefficient above is negative, so perturbations about axis three oscillate and remain bounded. Cyclically applying the same <linear stability of principal-axis rotation> calculation about axis one gives
$$
\ddot\omega_2
=-\Omega^2\frac{(I_2-I_1)(I_3-I_1)}{I_2I_3}\omega_2,
$$
so that rotation is also stable. About axis two, however,
$$
\ddot\omega_1
=+\Omega^2\frac{(I_2-I_1)(I_3-I_2)}{I_1I_3}\omega_1,
$$
which has an exponentially growing solution. Thus the <intermediate axis theorem> gives
$$
\boxed{\text{rotation about the smallest and largest inertia axes is stable, while rotation about the intermediate axis is unstable.}}
$$
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