I just had new hardwood flooring installed, including reconstructing my bottom 4 stairs to include false tread end caps to replace my previously carpet wrapped stairs. My next step is to have carpet installed by a different crew on July 2.

Here is my major design concern: The nosing of the new false treads sticks out pretty far, and I'm worried that when the carpet and 3/8" pad are installed, they won't be the same depth as the nose. So instead of getting the look of a wood tread with carpet in the middle, the wood nose is going to stick way out and be super obvious. Pictures below shows the new end caps, with my existing carpet that is about to be replaced. New carpet+pad depth isn't much thicker, unfortunately.

How can I work with what I have and make the false tread match up to the carpet more seamlessly? I know I shouldn't mess with the geometry of the stairs, but what are my options?

Thanks for your help. Carpet comes July 2, 2021.

Also I'm having a hard time finding info online about this, so if I'm using the totally wrong terminology, let me know.