My 65 GT (Metuchen build) was bucked on July 21st. Different plant than your car, but in this case it shouldn't matter. My GT is a power steering car and has the HCC-AW tag on the steering box. The date on my steering box tag is 5G13B (8 days prior to the car assembly beginning).
I've always understood the HCC-AX box to denote a manual steering car with the special handling package and / or GT package and the HCC-AW box to denote a power steering car (with or without the with the special handling package and / or GT package). There is no special GT steering box for a power steering equipped car. They simply received the same identical quick ratio box as all other power steering cars.
Functionally speaking, an AX and AW steering box both possess the same quicker 16:1 ratio vs. the 19:1 ratio on a regular manual steering box. The difference between the AW and AX steering box was in the preload specs. The torque required to turn the shaft is higher on the manual steering box (AX) as the physical forces on the manual box would be higher than on the power steering box (which is assisted by the hydraulic ram).
From the Shop Manual
HCC-AT Manual, 19:1, 4 5/8 Steering wheel turns, 4-5 in-lbs worm bearing preload, 9-10 in-lbs total preload
HCC-AX Special Manual, 16:1 3 3/4 Steering wheel turns, 4-5 in-lbs worm bearing preload, 9-10 in-lbs total preload (Special Handling and / or GT Package)
HCC-AW Power, 16:1, 3 3/4 Steering wheel turns, 3-4 in-lbs worm bearing preload, 8-9 in-lbs total preload (with or without the Special Handling and / or GT Package
PS - For what it's worth, I've personally never seen a factory GT with power steering that hasn't had a HCC-AW steering box installed.
Regards,
Ron