TS: Chet Baker's My Funny Valentine v.s. Julie London's Cry Me A River

There are plenty of other Julie London songs I prefer to "Cry Me A River" but "My Funny Valentine" seems like a career peak for the vocal side of Chet. This doesn't really answer the question though.

Did they ever sing each other's signature song?

London sang MFV, but I'm not sure if Baker sang CMAR.