Prefix complexity of x conditioned on prefix complexity of x
For plain complexity the assertion is true. One direction is clear: $K(x|K(x))\leq K(x)$ (easier to describe x from something than from nothing). In the other direction one can argue by contraposition. Suppose one could find $x$s with programs $q$s of length $l(q)$ arbitrarily shorter than $K(x)$. Set $K(x)-l(q) \equiv C$. In other words one can find $x$s with arbitrarily large corresponding $C$s. To reconstruct $x$ one needs to only know $q$ and $C$, which can be encoded in $l(q)+2\log(C)$ bits. Now $K(x)-l(q)-2\log(C) = C-2\log(C)$ which can be made arbitrarily large for sufficiently large $C$. Hence, we have obtained a description of $x$ much shorter than $K(x)$, a contradiction. This may be found as problem 43 in the following book (in Russian): https://www.mccme.ru/free-books/shen/kolmbook.pdf (Vereschagin, Uspensky, Shen, "Kolmogorov complexity and algorithmic randomness").