In any case, in your example, convterting from sRGB into ProPhoto, there can be a difference in the prints. If you print the converted ProPhoto with relative colorimetric, there will be no difference, since there could be no out of gamut colors. If you used perceptual rendering, then there would be a difference, since the entire gamut of ProPhoto would be compressed by a predetermined amount, even though there are no out of gamut colors.
/> Create any color in sRGB
/> Duplicate the file and convert the copy to ProPhoto RGB
Note: RGB numbers change, but Lab coordinates stay the same
/> Convert both files to your printer profile* using the Perceptual intent
(same setting for black point compensation, respectively)
/> measure the new RGB numbers, respectively
Finally, both RGB combos should be right the same. The Perceptual Intent has no idea about what the source space was. It just 'collects' the data from Profile Connection Space PCS Lab or XYZ, to apply its rigid non-linear transforms for compression into the target space.
*Epson generic, including all 6 common A2B and B2A tables. For control, a second test can be done, now using RelCol – which again yields identical RGB combos, but different compared to above test. As it was Ethan Hansen who explained this to me some time ago, I’m surprised to hear that his printer profiles wouldn’t contain respective LUTs…