Reading the IPNI database, I'll admit just skimming,
The botanical nomenclature is not "settled". It all depends on which "authorities you read.
Settled taxonomy - Carpinus koreana is nomen nudum - a misspelling, and therefore an incorrect name. An orthographic error. The correct spelling is Carpinus coreana.
Some authorities consider Carpinus coreana to be a valid species.
Some authorities consider the type specimens labelled Carpinus coreana to represent Carpinus turczaninowii period. NOT a subspecies of C. turczaninowii. These would label all Carpinus coreana simply as Carpinus turczaninowii, without any further descriptors.
Similarly, there are those that believe the herbarium specimens labelled Carpinus turczaninowii all really are Carpinus coreana, without the addition of any subspecies descriptions.
I do not know if the name Carpinus turczaninowii var. coreana has been legitimately (according to IPNI rules) published. But I have seen this name used often.
The point is, that all these names are equally correct except Carpinus koreana, that is the only name that is absolutely invalid.
So YES, to Carpinus coreana, Carpinus turczaninowii, and to Carpinus turczaninowii var coreana.
Until there is a monograph published doing a revision of the genus Carpinus, I doubt this will be settled beyond this level.