For the equation L = r x p, assuming that the implied rotation occurs around a central point.
Premise 1:
There is a force at all times directed from the point mass along the radius toward the centre of rotation (centripetal force).
Premise 2:
A change in the magnitude of radius is conducted by altering the magnitude of this force.
Premise 3:
There can be no component of this force perpendicular to the radius.
Premise 4:
In order to affect the magnitude of the component of momentum perpendicular to the radius, one must apply a parallel component of force (Newton’s first law).
Deduction:
A change in the magnitude of the radius cannot affect the magnitude of the component of momentum perpendicular to the radius.
Conclusion:
In the equation L = r x p, assuming that the implied rotation occurs around a central point, it is the magnitude of the component of momentum perpendicular to the radius that must be conserved when the magnitude of the radius changes.