Undergrad Rotation about two axes and angular momentum

[Kashmir]

I've a body having initial angular velocity at $t=0$ as shown. The axis shown are fixed in inertial space and initially match with the principal axis. I want to find the infinitesimal change at $t+\Delta t$ in the angular momentum along the $z$ axis.

I've seen the following approach which I don't understand:
One contribution to change in $L_z$ is due to rotation about y axis. This causes $L_x$ to rotate and hence a component $-L_x \Delta{_y}$ appears.

How do we know that $Lx$ will remain constant in magnitude? Also the actual motion won't be as is shown, in which the body simply goes around the y-axis while maintaining it's spin $L_x$

A similar method is used here by Kleppner and Kolenkow here

 

[anuttarasammyak]

May we expect Lz=0? At least periodic it seems.

 

[Kashmir]

May we expect Lz=0? At least periodic it seems.

Initially?

 

[anuttarasammyak]

Yes, and I assume $\mathbf{L}=(L_x,L_y,0)$ is conserved in later time.