With The Talos Principle by Croteam is one of the more interesting Linux test-cases. Croteam offers OpenGL and Vulkan renderer options on Windows to complement their Direct3D 11 renderer. This makes it interesting for comparison and as a reminder The Talos Principle was the first Vulkan game.
With the Direct3D 11 vs. OpenGL results we see the GL performance on both Windows and Linux to be much slower than Direct3D... But at least the OpenGL performance between Windows and Linux is about the game.
When looking at the Vulkan performance, the Linux Vulkan numbers weren't too far behind the Windows Vulkan performance on NVIDIA. The other good news is the Vulkan performance being faster than the OpenGL results, but still not as fast as the Direct3D 11 numbers.
When running The Talos Principle with ultra quality settings, the results were similar: OpenGL performance was lower than D3D11, but at least the OpenGL performance was the same between Windows and Linux. With Vulkan, the Windows vs. Linux results were close and for the GTX 1060/1080 were faster than the OpenGL performance.