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> A perpetually inverted pyramid would require that fertility rate shrink even further, until it approaches 0

That is incorrect.

A sub-replacement fertility rate will worsen an inverted pyramid situation, until the population eventually reaches 0

Given a constant fertility rate over time any inverted pyramid would eventually straighten out.

If you take a constant fertility rate of 1, you'll always have an inverted pyramid, until there's no more pyramid.

(I'm assuming that the life expectancy does not change significantly)