> finding a solution to something nobody else has managed to solve
In the 21st century, that is definitely something I'd bet on China for being able to do. Across all sorts of problems.
The solution is going to be paying people. No, more than that. No, keep going. Higher. Market-clearing price.
That's a capitalist solution; China has that option if they want it, but also they have those "reeducation" camps so it could get very Handmaid's Tale.
The question is how high that price is, and whether it is viable. We don't know. And secondly, whether or not the consequences will be workable for a cohesive society whether because women choose to have children only for financial reasons or get coerced into it for the same reasons.
It could get very dark very quickly if it turns out to be too hard to induce enough people who actually would want children under the right circumstances to choose to have them.