I'd also use "hasn't been", but for a different reason. It is probably still morning, because it isn't lunchtime yet. This is a subtle point, because we don't know when lunch starts, therefore we don't know whether Jake could still arrive "this morning", and this is what the usage depends on - as you say.
I'd call it a bad question - almost willfully insincere, unless I'm missing something. (Though they say not to attribute to malice what could be due to simple ignorance or stupidity).
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