Possibly. Unless the game can use more than 4 cores or the game happens to be extremely gpu dependent. The 1080 is usually for 1440p/4k, the lower the resolution the less work the gpu has to do. Pairing a powerful gpu with a lower resolution generally won't use it to its' full advantage. The only time a 1080 might be useful at 1080p is if looking to run full graphics (very high, extreme etc whatever the game calls it) and pushing for higher fps for 120-144hz monitors. In that case intel's higher clock speeds and ipc will generally be the best bet.
If it's a 60hz monitor at 1080p resolution then what you may experience won't necessarily be a bottleneck in terms of the cpu being an issue but simply too much gpu for the given refresh rate and resolution. 1080p is still widely used but compared to a few years ago where it was pretty much the standard, 1440p and 4k are becoming more common so they need stronger gpu's to drive them. Similar to how 720p was to 1080p, it didn't require as strong of a gpu.