"Linear curve" in most commercial raw converters are usually still not linear, a common thing they do is to clip away the last bit of highlights, and from the Capture One example above it seems like it does exactly that. It's prbably recoverable by just lowering exposure though.
I did a test yesterday with C1 to check if the ICC profile is applied before or after exposure adjustment (and curve), and it's applied after. I kind of new this already as you can export a TIF with the camera ICC, meaning that the ICC must be applied in the end. Phase One's ICC profiles has hue twists by the way, just as Adobe's DCPs typically have. I think Phocus own profile format is wihtout hue twists, but I haven't verified it for sure.
Anyway, this is a weakness of the ICC format, with Adobe DCP you can apply one correction at the raw before exposure adjustments tonemapping etc, and then apply one after. The first is intented to correct color (and they recommend against hue twists there), the other is intented to apply a look (and generally contains hue twists, eg saturation increase in shadows as many like that).
If there is a static color shift in the darkest shadows, that could be corrected with a DCP profile (using hue twists in the first HueSatDelta correction table), but it's not possible to correct with C1's ICC. In any case there's noone that does it today, but it could be an interesting new development in camera profiles. If it's like voidshatter says that the cast varies over time it's not possible to correct anyway though.
RawTherapee supports C1 ICC files, but it's color pipeline is designed such that profiles should only correct for accuracy and look is up to the user using the raw converter tools (which is a flow I prefer for my own photography), this means there is only one profile application step and that is before any adjustments, so if you push a heavily underexposed file the hue twists in the ICC will make it all look wrong, so to use C1 ICCs and indeed Adobe DCPs with a "look" in RT with desired result you need to have a file which has the right exposure from the start. RT's own DCPs are without hue twists and designed for accuracy at D50.