So far Lucasfilm Ltd. has failed in attempts to stop Andrew Ainsworth from selling replica costumes over the Internet. Ainsworth sculpted the galactic warriors' white helmets for the original "Star Wars" film in 1977.
The High Court ruled in 2008 that the costumes were not works of art and not covered by British copyright law.
The Court of Appeal upheld that decision the following year.
Now Lucasfilm is asking the Supreme Court to rule on whether U.S. copyrights on "three-dimensional works" can be enforced in England.
The hearing began Monday and is set to last three days.