I think that would require a modern species to be compatible with a direct ancestor from the distant past, with generations of incompatible beings in between. Otherwise there is no "ring". But that seems 1) unlikely, 2) untestable.
Whoops, I guess you're right. I read about ring species long ago and I definitely misunderstood the concept, so I'm glad you corrected that. I was thinking all neighbors are compatible, but incompatible with the opposite side.
However, because of the impossibility of contact between the two temporal ends, I'm not sure what the value would be in thinking about a temporal ring species?
The kind of ring species you refer to would also be possible, although far less likely.
The ancestor species could form a geographical ring, and if the members are not too populous, nor too migratory, then mutations that cause local incompatibility would be selected out (because it makes it much harder to find a mate) whereas mutations that cause non-local incompatibility may persist (if the species is largely non-migratory, so non-local compatibility is not as valuable). Over time, the non-local incompatibilities can build up and create a ring species like the one you describe.
But something like this is far less likely than a ring species with incompatible ends. Because such "complete" rings can decay into incomplete rings through the extinction of a connecting subspecies, and similarly, incomplete rings can segment further, and then grow again as two separate ring species. But a complete ring, once broken, is very unlikely to re-form.
Interesting in that in population spread, expanding rings aren't entirely improbable - if a population has a tendency to exhaust resources when it expands into an area, its spread could naturally form loops around the edge of depleted, unpopulated areas. It would be interesting if there could be cases where a species expanded like that in the past, and evolved under that kind of closed ring pressure. Even if later that ring broke, how would that ancestry show up in the surviving species' taxonomy?
The evolutionary biologist Richard Dawkins remarks that ring species "are only showing us in the spatial dimension something that must always happen in the time dimension".
While the term applies more broadly, it's called a “ring” from geographic examples where the ring is closed geographically, with adjacent/overlapping incompatible populations, but where there is a literal ring of geography wherein adjacent populations interbreed continuously except the two adjacent incompatible ends.
I think that would require a modern species to be compatible with a direct ancestor from the distant past, with generations of incompatible beings in between. Otherwise there is no "ring". But that seems 1) unlikely, 2) untestable.