When using the '#'-character in a field, JabRef
complains that 'The # character is not allowed in
BibTeX fields'. However, the Bibtex documentation
suggests that the month field could be used in the
following way:
month = jul # "~4,"
(Example from Oren Patashnik, BIBTEXing)
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In JabRef you have to write this as "#jul#~4", which would
be stored in the way you say. This is to avoid ambiguous
situations, since JabRef can't reliably detect what is a
string label and what is not.