All methods declared in an interface are inherently public. So any implementing classes must also declare the interface methods as public. Since you can not override a method with a more restrictive access modifier, then all method overrides in any subclasses must also be public.
This is the objective of an interface, it is a contract so that all implementing classes are guaranteed to provide public implementations for the interface methods.
Abstract class can leave the implementation of the interface it declares that implements but can't reduce its visibility. The first concrete class must implement that interface methods (must with public visibility) + abstract class methods (which are abstract) with the same visibility or more public visibility.
In case of public visibility (as default of interface methods), no other reduced visibility is possible, obviously.