> Why do you believe inverted pyramids don't last?

Death. And there are trivial ways to keep fertility at replacement level and not negative, on par with most other 1st-world countries.

A perpetually inverted pyramid would require that fertility rate shrink even further, until it approaches 0 - do we have a reason to believe this will happen?

Notwithstanding, in 100 years the entire global population growth is projected to stagnate, which will make this a moot point. No one will be able to rely on immigration to grow perpetually.