Here I have a model which I am sculpting. In Sculpt mode it is smooth, while in the Object mode it is not. There are no modifiers or anything, so what could be the problem ? Here is the file for a part of the object:
Custom split normals data
Your model has a custom split normal data (probably due to a normal edit modifier that has been applied or because the mesh was made with another software...) that keep some face shading in that way.
If you don't need the data, you can clear it to remove the informations, or you can just disable auto smooth to make the split normals data not visible.