A good question.
Once I've posted on another board the same question after having read
"The Art of the Japanese Sword as taught by the experts" by
Kunihira Kawachi and Masao Manabe.
At page 61 Kunihira Kawachi clearly shows and states how to apply
the oil to Nakago too, with appropriate amount (not too much but
not too few) and with a different clothe then the one used for the
blade. The sword seems one of theirs, so newly made.
Avoiding to mention the usuall "I know everything" that addressed
me to a link on how maintain a Japanese sword where is shown how
to use the remnant of oil on you fingers to very sligtlhy oil an old
nakago, the thread resulted in an agreement (?) that old nakago with
already well-established patina should be preserved with a slight oiling
as the "finger remnants" matter, but that new or not so old Nakago should
be left get rusted and patinated.
At the end of the matter : if the tang is old enough to risk a damage to
important features as yasurime or Mei or other inscriptions/details with
potential further corrosion, let's apply a very small amount of oil.
If the Tang is new or still healty, avoid oil.
I'm just quoting the result of that thread, not stating my personal
opinions or challenging somebodyelse one. :D