Is this sentence correct?

Dear marcuccio,

I'm happy to respond.

The closer/further rule gets tricky when we invert the word order.

Think about it this way. The first auxiliary verb can hop around to all sorts of position in the sentence. The rest of the verb has a definitive location. Let's say that the location of the main part of the verb, the "meaning" part of the verb, is at the "been seen." The "has/have" would be there if we hadn't changed the word order. Then the sentence would be
In several days, the mother or the children have not been seen.
or, more naturally,
In several days, neither the mother nor the children have been seen.
In either case, of course, we need the plural verb, "have."
When we are thinking about closer/further, we have to judge with respect to the main verb location, the location of the "meaning" part of the verb, not the unusual position of the auxiliary verb.

Another way to say the same thing is: changing the word order, moving the leading auxiliary verb around, does NOT change which verb we would use. We always use exactly the same verb that we would have used if the whole verb were together in one place.

Does all this make sense?
Mike