Clarification regarding Principal axes in rigid-body motion

Question: We need to find, the angular momentum of the assembly, about the Center of mass.

As per Kleppner and Kolenkow, the general Expression for $vec{L}$ about any point is:
$$vec{L_{p}}=I_1vecomega_1+I_2vecomega_2+I_3vecomega_3$$
where $I_1,I_2,I_3$ are moments of inertia about the principal axes. As far as I understand, these "principal axes" pass through though the point P.

However, an (although excellent) blog post:https://crazycosmos.wordpress.com/2017/12/08/rigid-body-motion-the-iit-jee-saga-i/, under the heading truth of part A, selects the principal axes in such a way that two of them dont pass through the center of mass!

Am I incorrect in my understanding that all 3 principal axes must pass through the point? Was there any reason to chose the principal axes (the 2 except the axis symmetry), that dont pass through the point?