In the actual process, the stuff with the lowest melting point, the white cast iron, would stay liquid.
The stuff with the highest melting point, the WI, would precipitate. Any carbon would rapidly diffuse out into the atmosphere.
Mild steel has a much higher melting temp than CI - by hundreds of degrees F. Iron and mild steel are just a smidge apart on that scale. Your temperature controls going to be that tight?
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also...
Tell me again how you plan to get the ferrosilicate into it? Ain't none in mild steel, you know. Plenty of it in white cast iron, though.
With no source of ferrosilicate, you'll only decarburise the mild steel into milder steel. Won't be "pure iron" because it will have manganese in it. Won't be WI because it won't have ferrosilicate.