OSPF - default-information originate : multiple routers

I have a very simple topology:

I receive a default route via iBGP from R1. On both R2 and R3 I have default-information originate configured. On R4 I only see one default route in the OSPF database; either to R2 or R3. When I shutdown one of the links from R2 / R3 to R1, I see the other default route appear in the OSPF database.

My real topology is a bit more complex, so is it possible to have both default routes appear on R4 without default-information originate always? I want the default route being advertised on R2 / R3 to be reliant on the default route from R1 existing in the RIB.

Thanks

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