As the fluid drains from the lines, it might not be pulling air into the reservoir at the top of the system, but air is flowing into the lines from the bottom as more fluid drains out. I don't really know how long it would take for a bubble to rise if the line were to be capped, but I'd be looking for rubber plugs from a hardware store. I imagine limiting the amount of fluid that leaks out would limit the migration of air.
Lowes and Home Depot don't sell little tapered plugs- around here we have McLendons that still carries stuff like that.
https://www.amazon.com/Assorted-Sizes-Solid-Rubber-Stoppers/dp/B07MZJG3FC/ref=asc_df_B07MZJG3FC/?tag=hyprod-20&linkCode=df0&hvadid=312135717511&hvpos=&hvnetw=g&hvrand=827552023521595362&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9052871&hvtargid=pla-652514712186&psc=1
If the link works, it's just meant to illustrate what I'm talking about, they look like they're the wrong size for this particular use.