How to stop advertising a prefix to a BGP peer if the route to the prefix goes down?

I have a prefix advertised out of two different sites. The prefix is only in use at site A. There is a p2p link between the sites. Site A uses that p2p link as it's primary default route.

If that p2p link goes down, how to do I get the router at site B to stop advertising site A's prefix so site A will be reachable via it's backup ISP.

Link to simple diagram

I have OSPF running between R1 and R2.

If I shutdown the p2p link, R1 keeps on advertising SiteA prefix to both ISPs. I don't have a static null route for SiteA prefix on R2. I was hoping that when the OSPF learned route dropped, it would stop advertising it via BGP as well.

Would this work if I dropped OSPF and used iBGP instead?

Sorry if I boiled this down too much. Please let me know if I can clarify anything.