For Kolmogorov complexities $\hspace{.02 in}K$ induced by essentially-optimal description languages,
does there exist an integer $c$ such that for all positive integers $n$,
there exists a string $x$ such that $\;\;\; n \: < \: K(x) \: < \: n\hspace{-0.04 in}+\hspace{-0.03 in}c \;\;\;\;$?
Unlike the question I was inspired by, this question's answer is robust against the choice of $\hspace{.02 in}K\hspace{-0.02 in}$, since by definition $L_{\hspace{.03 in}0}$ is an "essentially-optimal description language" if and only if
it is a computable partial function from $\{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.03 in}\}^{\hspace{-0.03 in}*}$ to itself such that for all
computable partial functions $L_{\hspace{.02 in}1}$ from $\{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.03 in}\}^{\hspace{-0.03 in}*}$ to itself, there exists an
integer $c$ and a computable (total) function $\: f : \{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.03 in}\}^{\hspace{-0.03 in}*} \to \{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.03 in}\}^{\hspace{-0.03 in}*} \:$ such that
for all strings $x$, $\;\;\; \operatorname{length}(\hspace{.05 in}f(x)) \: < \: \operatorname{length}(x)+c \;\;\;$ and $\; L_{\hspace{.03 in}0}\hspace{-0.02 in}(\hspace{.05 in}f(x)) = L_{\hspace{.02 in}1}\hspace{-0.02 in}(x) \:\:$.