As a general rule, true soap (whether made with KOH OR NOH) isn't great for hair. The pH is too high and it can damage the hair. That said, my hair isn't horrible with a CP 'shampoo' bar that I make, although I prefer to use my liquid surf based shampoo. Some people have hair/scalp that does well with true soap, but not all, or even most. If you follow with an ACV rinse it helps a lot.
If you add an acid to liquid soap, well I'm not sure how low you could actually get the pH without it doing something wonky to the integrity of the soap itself. I haven't tried to lower my pH below 9, so I can't tell you if it would work. You could use a CP recipe for LS, but you want different properties in a LS than in something solid, because hardness is no longer an issue, or even a factor at all. The book "Natural Liquid Soaps" has recipes for liquid soap shampoos in it, and is worth a read. Oh, and silk is awesome in shampoos! Unless you have fine, flyaway hair, then it is better to use something with a higher molecular weight like wheat or oat protein.
You should note, though, that if you do lower the pH below 9, and if you add any plant extracts of any kind, you should add a preservative. Citric acid and vit C don't act as full preservatives, and you could still get icky stuff growing inside. Personally, I usually use Geogard Ultra as my preservative. It is EcoCert certified, and doesn't have 'scary' chemicals in it. It's as close to a natural preservative (that works) as you can buy, really. There is also Naticide, which is extracted from essential oils, but it is quite pricy and I've heard it fails challenge tests for some.