No it wasn't. Khronos' own slides on Vulkan disprove this (slide 6) [1], saying they began the effort in June 2014. Apple announced Mantle to the public that month.
In the same presentation as you linked, on slide 9, you can see Apple was a part of the Vulkan effort. The subsequently withdrew from Vulkan, before releasing Metal.
WWDC 2014 in June had the 'Zen Garden' demo running Unreal 4 on the Metal API [1], and it was released to the app store in September [2]. In a different session, CryEngine was shown running Metal on an iPad [3]. That's two high-profile game engines already using the API. A limited release perhaps, but I count that as a significant head-start, all the while Khronos was just organizing glNext.
