Yes. In contrast to Mozzarella, Champagne, and many others which are generic products in the USA, and protected trademarks in their region of origin.
I didn't realise the trademark holder is a US company.
Only mozzarella di bufala campana has PDO, even in Italy there are non-PDO, generic mozzarella cheeses produced outside the region.
Champagne is absolutely not a "generic" product in the US.
While some California wines were and still are sold as "California Champagne", there was a court decision back in 2006 which affirmed the same protected trademark for Champagne as in the rest of the world, with the caveat that wineries already using the "Champagne" name would be grandfathered in and permitted to continue doing so.
A bottle of Champagne made in the US will be labeled with its origin, and no one can start a winery and start selling Champagne in the US. That's not the same thing as "generic".
To clarify, Uggs originated in Australia and are a generic term here. Deckers owns a trademark outside Australia and is enforcing it against people in Australia.
It would be like if a US company registered Champagne as a trademark then enforced it against French companies selling it.