How about:

$HOHOH{'Alphakey2'}{'Betakey2'} = $value;
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However, I have a feeling this is not what you want to do. For instance, this precludes you later from setting a value for 'Alphakey2', 'Betakey2', 'Gammakey2'. If that's what you want to do, you'll probably want to use the comma operator like this:

$hash{$alpha,$beta,$gamma} = $value;
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The comma operator here is syntactic sugar for "$alpha$;$beta$;$gamma", so you are only creating one hash but with a special key. To not include a gamma component, just omit it:

$hash{$alpha,$beta} = $value;
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Update: It just occurred to me that maybe you want to do this:

$HOHOH{'Alphakey2'}{'Betakey2'}{undef} = $value

which preserves the HOHOH structure.