In Carinthia, Bolzano Airport's own brand, Skyalps, is currently being discussed for possible flights to Frankfurt am Main and Munich. On a charter basis, this should serve the hub connections announced by Lilihill.
Skyalps was launched some time ago following the privatization of Bolzano Airport. The operational execution of the flights is currently being carried out by the Maltese Luxwing, but that should change shortly, because you will receive your own AOC and your own operating license.
Skyalps already has occasional out-of-base operations. This could also be implemented for Klagenfurt. However, there is no official confirmation from either Bolzano Airport's own brand or from the Lilihill Group. In December 2022 you want further details about the planned flights to Frankfurt and/or Munich. Without a Lufthansa codeshare, however, it would not be a real hub connection.