Do dead bodies pose a problem for biological approaches to personal identity?

Part of the appeal of the biological approach to personal identity is that it does not have to countenance spatially coincident entities. But if the termination thesis is correct and the organism ceases to exist at death, then it appears that the corpse is a dead body that earlier was a living body and distinct from but spatially coincident with the organism. If the organism is identified with the body, then the unwelcome spatial coincidence could perhaps be avoided. It is argued that such an identification would be a mistake. A living organism has a different partwhole relationship and persistence conditions than the alleged body. A case will be made that the concept 'human body' is a conceptual mess, vague in an unprincipled manner, and that an eliminativist stance towards dead bodies is the appropriate response.