Right. The goal here was to most closely create a rendering in sRGB that would render to a printer how ProPhoto would render without clipping to sRGB first which produces a less faithful rendering due to RGB matrix math clipping.
Yes, I see the point of your suggested approach, and no doubt it can be useful right in the context as described in post #49 by FranciscoDisilvestro.
But then it elevates the conversion from ProPhotoRGB to the printer profile - and its presumably more advanced gamut mapping / clipping algorithm - to a kind of general reference. Below is an example where I think it doesn't work out.
… since we were also talking about Fuji Frontier Printer along this thread:
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Image #01: was kept in ProPhotoRGB with out-of-sRGB marks, just to illustrate that there is a bunch out-of-sRGB colors/pixels to deal with.
Image #02: was obtained by straight conversion from ProPhotoRGB to sRGB. As expected, and visible on my screen, the Hue gets slightly shifted towards yellow/orange, the red channel is clipped and fine details are getting somewhat blurred. That's not ideal, but we will see this is not the worst case.
Image #03: was obtained by ReCol conversion from ProPhotoRGB to one of the Frontier Printer media profiles which Fuji Europe offers
here for the Frontier Printer in so-called sRGB-mode. Next, the image was converted RelCol to sRGB to ensure the comparability with the previous image #02. This second conversion does not do a lot. The deterioration of the image happens with the first conversion.
Conclusion: whatever RelCol-gamut-clipping-algorithm this printer profile uses, it is apparently worse here than the simple matrix math clipping. Image #03 is quite dark and the fine details got largely blurred. Further, I also tried one of the Fuji Frontier profiles from Dry Creek Photo, and the results are not really better.
Image #04 provides a cross check. The sRGB image #02 is converted RelCol to the Frontier Printer profile. Again the results are better than bumping into the printer profile directly with the ProPhotoRGB colors.
Image #05 shows the "semi-color-management approach" which Fuji once suggested in a corresponding paper (which is unfortunately not available on their site anymore). The image in sRGB, #02, is soft-proofed to the printer profile with Preserve RGB Numbers enabled. It was suggested to edit the sRGB image under this soft-proof before sending it to the lab. This approach may sound strange, but it is not wrong, under the premise a) that the printer's gamut is somewhat close to sRGB, and b) that the printer/lab do not run any ICC-type color space conversion before feeding the RGB data into the printer.