Converting from sRGB to ProPhoto y viceversa on Photoshop

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This is just academic:
I want to find a ciclycal way to convert from sRGB to ProPhoto and viceversa on Photoshop. I notice that no matter that the folowing images (1) (from Is Your Browser Color-Managed?) looks diferent inside of PS, the browser renders both identical (2) and that’s the goal of the ICC profile, I know. But, when I try to change de color profile of the proPhoto one to sRGB (3) on PS (with “convert to profile…” and “assign profile”) that doesn’t work and actually looks the same on PS and of course renders different on the browser. And with the same resultant image I tried to convert it back to proPhoto (4) and that doesn’t work either. There is a cyclical way to convert a image between sRGB and Prophoto and making them to looks the same on the browser?
Thanks in advance.

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