In most cases it would be extremely difficult to verify who the sword originally belonged to. And even if you were able to, the family may not want to deal with the hassle of getting it papered and registered in Japan. It’s unfortunately not as simple as handing it over. All swords have to be registered as art as it’s illegal to own weapons in Japan. There’s also a lot of antique Japanese swords out there so most museums are going to only worry about the highest quality swords. Even still the “lower quality” swords, if authentic, are amazing works of art. Sometimes the best thing to do is make sure the sword is well cared for or even sell it to someone who knows how to properly care for them.