Thanks for trying to reassure me, but I'm not even going to try it. My hair hates oil in any form, at any time, under any circumstances. If there is oil in the solution, it's going to contact my hair, and therein lies the problem--doesn't matter what the solution is. I don't know how BKT solution is any different from other solutions in that aspect.
I'm not sure I buy the whole protein sensitivity thing--I know...another of my unpopular opinions. But, it simply doesn't make any sense to me chemically--what possibly could protein do to your hair--which is mostly protein anyway???? Perhaps it alters the position of the scales on the hair shaft. I haven't read about CG so, maybe that will clarify it for me.
Oil, on the other hand, affects my very fine and very porous hair shaft, which soaks it up like a dry sponge, resulting in hair strands that are gummy, limp, stringy--almost wet in feel and appearance. I can always tell when there is oil in something, even without prior knowledge of the oil content.
I think each of us has different expectations for our hairstyles, which is why we claim sensitivity to one product ingredient or another. For instance, I don't like the well-defined, big, thick, clumpy curls some women favor, so I wouldn't use a product or a method that promoted that result. I like a big, wild lion's mane of curls. At one time, I thought my hair was "sensitive" to something in Curlisto Structura Lotion because when I used it, my hair looked like %@#*, so I pushed my huge bottle to the back of the cabinet and left it there...then I tried it again a year later, and it makes my hair look amazing. I also love how hairspray makes my curls look, but I'm sure some others would be mortified by their perceived level of stiffness and "dryness" of my curls and because they may hate the look or feel, would say my hair is "sensitive" to <insert substance here>.
Just my humble opinion.