CMYK or Gray Gamma 2.2 for gray scale images in a document with CMYK color images

Hi,

I want to combine color photos (CMYK) and gray scale photos in the same document using Illustrator and afterwards export to pdf, but I don't know what profile I should use for the gray scale images (CMYK or Gray Gamma 2.2). The images need to be printed for a book.

My workflow is:

1) Acquire photos with Capture One 20

2) Export from Capture One to tif (Adobe RGB 1998)

3) Adjust images with Photoshop, including conversion to CMYK profile

4) Adjust separate (gray scale) images with Photoshop, conversion to either Gray Gamma 2.2 or CMYK (here is my question)

5) Import color and gray scale images to Illustrator

6) Export to pdf

When I convert gray scale images in Photoshop to Gray Gamma 2.2, and import them to Illustrator to my other (color CMYK images), the gray scale images appear very pale, also after the export to pdf.

When I convert the same gray scale images in Photoshop to CMYK and import them to Illustrator, the appearance of the gray scale images is consistent, they look the same in Photoshop, Illustrator and pdf. For pdf export, I use Illustrator standard (PDF/X-4:2010). The result is fine, but is it a problem if gray scale images have a CMYK profile, so are they usually printed without a a color tint? (e.g. a bit blue or green instead of a nice gray scale)

Best,

Ankylo