This is very cool and quite exciting, Na'Tosha Bard, a developer for Unity has teased a little shot of Unity + Vulkan. Looks like it's running on Ubuntu too.
Note: For the record, this doesn't mean Vulkan is coming any time soon, but it is being worked on.
Na'Tosha later replied to me to state that Levi from Unity is the one doing the Vulkan work, as Na'Tosha is working on SDL for Unity currently.
Not surprising really, since Vulkan support is officially listed as in-development on the Unity Roadmap.