Cooldown Recovery Support in Tawhoa's Felling

Cooldown Recovery Support doesn't support Tawhoa's Chosen, the triggered skill on Tawhoa's Felling.

I've checked a handful of other uniques with triggered skills that have cooldowns, and it does support each and every one of them (though it claims to not support Lightning Warp from Moonbender's Wing, but still does).

Some examples of Uniques and their Skills that I tested to work:
Moonbender's Wing - Lightning Warp
Ngamahu's Flame - Molten Burst
Limbsplit - Gore Shockwave
Kongming's Stratagem - Fog of War
Eber's Unification - Void Gaze
The Dancing Dervish - Manifest Dancing Dervishes

I even checked Kitava's Thirst and the cooldown it forces on socketed skill gems, it applies there.