There's a lot more nuance to the situation than that... different GPUs will have different performance footprints on Vulkan, and the actual ways these engines use Vulkan can differ greatly.
Generally, Vulkan yields better performance than directly-equivalent DirectX code. Even since the early days of commercial Vulkan use (2016), the performance uplift has been a notable upside. However, Vulkan code is generally more difficult to write than DirectX (or even OpenGL and Metal, AFAIK), which is part of the price you pay for portability. High-level libraries were expected to paste over these issues, but this infrastructure is still in it's infancy. Ironically, one of the best uses for Vulkan right now is acting as a sort of intermediate representation for DirectX/Metal API calls.
Vulkan vs what? DX9 or DX11 or DX12. Also it really matters how well you wrote your Vulkan layer. If you write a good DX12 layer and a good Vulkan layer I think they will be comparable in perf - The bottleneck will be the GPU and/or driver. So it also depends on how well the driver writers did.
Generally, Vulkan yields better performance than directly-equivalent DirectX code. Even since the early days of commercial Vulkan use (2016), the performance uplift has been a notable upside. However, Vulkan code is generally more difficult to write than DirectX (or even OpenGL and Metal, AFAIK), which is part of the price you pay for portability. High-level libraries were expected to paste over these issues, but this infrastructure is still in it's infancy. Ironically, one of the best uses for Vulkan right now is acting as a sort of intermediate representation for DirectX/Metal API calls.