Is angular momentum perpendicular to fixed axis of rotation constant?

[TahirMaqbool]

TL;DR Summary

Does the component of angular momentum perpendicular to the fixed axis of rotation change in direction or magnitude?

So my book states torques perpendicular to the fixed axis of rotation tend to tilt the axis , however we assume sufficient restraints exist so these torques are simply ignored.
It follows that angular momentum perpendicular to axis remians constant.
(See image )

My question is that if a rod is hinged at one of its ends and allows to rotate, wouldn't angular momentum perpendicular to axis change in direction at each point?
See image below.

 

[Lnewqban]

Are those “torques perpendicular to the fixed axis of rotation” also contained in the plane that is “perpendicular to the fixed axis of rotation”?