To convert an RGB file to a target CMYK space, open the image file in Photoshop, then go to Edit > Convert to Profile. Once there, select your target profile (a copy of which should be in the [Your Computer] > Library > ColorSync > Profiles folder, or else it will not appear in the pulldown menu).

I would not recommend to convert RGB vector content to CMYK in Photoshop because it is more intuitive. You will lose the resolution independence of the vectors and give it a specific resolution. The custom CMYK you mentioned is based on outdated technology from the early years of Photoshop.

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Убαжа ևтθዶ կዓሮйуց оሃеበКаςዋжሁկ апашθፆጀኁеվΛиሒы я
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As a "test", I copied the AI RGB file and pasted it into a Photoshop Adobe RGB document, On-screen the AI file matched the PSD file side-by-side (so far). In Photoshop, I went to Mode>CMYK and the colors "slightly" shifted to a less saturated version (as expected), but totally usable for print (perceptually the same for most people without ghk9rW.
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  • convert rgb to cmyk without losing color photoshop