Here's proof of concept. I really like the simplicity of this demonstration to explain what we mean when we say the NH3 is still there and it's still a gas.
I took tank water and added ammonia drops to ~4ppm.
One part was untreated, one part I added 4x dose Prime and mixed. Then I poured the mixed 40mL into 50mL tubes, covered the tops with a paper towel, put a couple of seachem NH3 films on the paper towel and covered the whole thing in parafilm. The water never touched the paper towel, the NH3 films are simply responding to the gases over the water.
Top pic is after 1-2 hours. Bottom pic is overnight.
Left to right Tubes: Control - no ammonia, Ammonia untreated, Ammonia + 4x Prime, (and as bonus the rightmost is with ClorAm-X)
I did this because it was faster than waiting for NH3 to diffuse out of one sample of water and into another. But this strongly suggests that's exactly what would happen, whether you do or don't treat with Prime (or likely other products).