Suppose Company "A" registered the trademark "John Doe" with the EUIPO.
Company "A" has declared bankruptcy and been liquefied, but the trademark was retained by Company "A"'s founder.
During this time, a local company began to illegally use the trademark "John Doe" in a singular jurisdiction, but was not sued by the trademark's owner, due to general indifference.
The trademark "John Doe" has just expired, and its owner has no intent to use it in the future.
Could a third party then come along and re-register the trademark "John Doe" (either in all EUIPO jurisdictions or all EUIPO jurisdictions except where the local company operates)? The question Can I file the same trademark as one that is cancelled? discusses how one cannot register a dead trademark if the owner intends to continue using it, etc. However, can the trademark also not be registered by a third party if another company has begun to unlawfully use it prior to its expiration?