How to avoid Pure Black when converting RGB images to CMYK

After getting a coffee table book printed and being aghast at how incredibly black some of the very bright images printed, I have been searching for the reason why. Ariel posted, some years ago... The most important point is: don't use pure black on images. Never. The result could be a disaster, specially if the images has a lot of dark areas. Pure black is important for vectors and text, but it's a bad choice for convert an image from RGB to CMYK.
I believe that quote above answers it. Since the printer requested CMYK, in my care to get it right, while grading I soft proofed my RGB images in CMYK and then exported them while converting them to CMYK (embedding FOGRA39). Then I linked them to the document because there were so many. In CorelDraw 2019 color management, the 'Preserve Pure Black' is checked by default. So I assume all the images got converted as Pure Black instead of Rich Black? Am I right in making that assumption?

And this mistake was not visible to my eye when viewing the converted images on screen! They looked a shade darker, but nothing like they printed at the litho.

To fix this, should I go back to the original RGB images and leave conversion to CMYK until the output to PDF? Will this avoid the 'Pure Black' trap?

Indeed, how does one convert images to CMYK safely? Just turn off 'Preserve Pure Black' for each conversion, and back on again for the text?

I also notice in the Soft Proofing dialogue a box called 'Preserve CMYK numbers'.  Should that be checked?

Any help with this would be gratefully appreciated!