converting edited 4K footage to 1080p with Compressor 4

You hear quite a bit about how 4K footage can be downconverted to 1080p...and that because of the downsampling form a higher res original, this converted 1080p footage would look sharper than 1080p footage originally shot in a camera.

My first question: I assume Compressor 4 can handle a conversion from 4K to 1080p (please correct me if I am wrong?)

And secondly, has anyone done a test comparing regular 1080p camera footage to 4K downsampled 1080p footage (via Compressor 4)?

I am trying to determine whether there is some validity to the claims of the superiority of 4K downsampled footage...and whether Compressor 4 can handle that?

(A side observation here...I have used Compressor to convert 1080p ProRes footage to 720p (h264) for web use, but it comes out soft...and I have to use the Sharpen Filter in Compressor...on the setting of "3" to deliver a sharpened image. Wondering if the same process would needed to be down when converting 4K to 1080p ProRes)? Or maybe the sharpness holds up when going Pro Res to Pro Res...versus ProRes to h264.)

Thanks for any advice you can share,

John