Hi,
The best thing to do is remove the eggs. Leaving them in there can encourage nesting. Make sure she has nothing she can sit in or go inside of – no birdie huts or tents, nest, box, etc.
There is always a concern with housing different species together, especially when there is a difference in beak size. “Coexisting pretty good” is not ideal. A harmless squabble can turn deadly without the conure meaning to harm the budgie, but a budgie is no match for a conure beak. And these are species that come from different parts of the world, so were never meant to coexist.
Thank you for asking Lafeber,
Brenda