Bird Cage Theater + Conditional Control Points

How does Bird Cage Theater interact with cards like Desolation Row or Max Baine that have a trait that generates conditional control points? Do those control points immediately come back so long as the condition is met? If not, is there any way to get the control point back?

Relevant text:

Bird Cage Theater
Controller Noon Job, Boot: Mark an opposing dude with 1 or more control points. If successful, remove all control points from the mark and this deed gains a permanent +1 control point and +1 production.

Desolation Row
Your dudes with 4 or more bounty are worth +1 control point.

Max Baine
Max is worth 1 control point while you control 2 or more Ranches.

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Hello.

Control points granted by traits will return once the Bird Cage Theatre’s Noon Job action is completely resolved.

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