Hi @clement,
Ideally it would all “just work”, and the reflections (the object’s normals) would be updated according to the new geometry. You can get most of the way there by assigning the same texture also to the Bump / Normal slot:
I created a YouTrack item for the issue: https://mcneel.myjetbrains.com/youtrack/issue/RH-70056/Display-Displacement-texture-doesnt-modify-object-normals
-David