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Liverpool didn't fire Jurgen Klopp because Klopp decided to retire himself so I think Liverpool still has a good relationship with Klopp. In my opinion, there is a possibility that Arne Slot will be selected because of Klopp's suggestion. For the first stage, Liverpool cannot hope to achieve good results in the UCL and EPL. Arne Slot also requires tactical adaptation to each Liverpool player. Every coach has a different way even though they have the same philosophy. In modern football I don't think big names make much of an impact. A coach must be able to adapt to various variations of tactics and watch replays of each opponent's game. I think a flexible coach is the coach Liverpool needs right now.
Usually there will be an impact after a change of coach at a club, Arne Slot inherited an established squad from Jurgen Klopp, he has a big responsibility to manage Liverpool like Jurgen Klopp once did. I didn't expect much from Arne Slot first season with Liverpool, he needed time to develop a strategy and get the players used to the tactics he used.
Liverpool who will appear in the UCL, will have a busy schedule, Arne Slot must be able to make Liverpool perform optimally in different competitions to get support and attention from club management for their performance. Arne Slot's experience with the previous team means he doesn't need too much time to improve Liverpool quality, he just needs to maintain Liverpool quality to provide competition in the EPL trophy hunt and reach the UCL semifinals in his first season.