I am not saying that CX formally releases award inventory for any OW partner. However,as an AA EXP I have gotten award availability released on all my First/Business awards close to date of travel. I avoid Chinese new year holiday and a few peak days just to be careful.
I also travel 10 to 20 paid discounted F cabin longhaul trips on CX each year and again never had a problem - CX release seats (for discounted F) for AA EXP even if they show zero availability on CRS.
YMMV.