On the marketting blurb web page, the only mention of damping is this:
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No other adjustments are necessary. Through blind testing with a variety of riders, Cane Creek decided that complex and wide-ranging damper adjustments simply aren’t needed when considering the minimal travel and primary use case of realistic gravel riding. Invert is everything a rider needs, and nothing they don’t.
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So, it doesn't explicitly say whether or not there is damping, just that it is not adjustable.
In regard to the lockout feature, it says this:
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Invert CS comes with 40mm of fork travel and a Climb Switch, that when clicked, dramatically reduces the amount of fork movement for pedaling out of the saddle,
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This implies that the lockout doesn't physically lock the sliders. However, if there was air damping, then the damping would occur as air passed through a valve between two air chambers. If the valve was locked shut (by the Climb Switch), then one (or both) of the chambers would act as a small volume air spring, limiting the amount of fork movement.
I think it is still possible that there is a (non-adjustable) air damper, and that the lockout switch locks the damping valve shut.