Very strange! Do you have a link to some images?
IE does support embedded profiles, but does not use the host's display profile:
Firefox supports it's host's display profile if you set gfx.color_management.display_profile to your current system's installed display profile - specify the full path. You can also turn on v4 support with gfx.color_management.enablev4, matching IE's behavior. Point firefox to about:config to change these....
I think that's what I said for IE - it supports image profiles, but ignores display profiles. It apparently "passes" the test on the link you give, as that test doesn't test whether the display profile is used.
For Firefox, I don't that's quite right. Colour management defaults "on", and you don't have to set gfx.color_management.display_profile unless the system default isn't the correct profile. If that parameter is null (the default) than Firefox uses the Windows system default (which is normally what one would want). Yes, you need to turn on v4 support if you want it, but lots of programs don't support v4 profiles (including Lightroom) and I would recommend anyone to use only v2 profiles for the foreseeable future. The only about:config parameter in Lightroom that I alter is gfx.color_management.mode, which I change from the default of 2 to 1. By default (if the parameter is 2), Firefox doesn't colour manage graphics with no embedded profiles. This is rather annoying on a wide-gamut monitor, as virtually all graphics objects are sRGB, and without colour management are too saturated - which of course is the core of this thread!
Curiously, Chrome gets this one right (it colour-manges untagged graphics, i.e. graphics without embedded profiles), but even then you have to turn on colour management in Chrome by starting it with command line switch “--enable-monitor-profile”.
Edited: sorry, you asked me for a link to images. I can't think of anything suitable, although I could make some screen captures perhaps. I've figured out what the various browsers do by trying them with various images in different colour spaces, both with embedded profiles and without, and on two different monitors: one wide-gamut and one approximately sRGB. In combination, that shows pretty well what isn't working.
One benefit of having two monitors with markedly different colour gamuts: colour management problems are immediately very, very obvious!