Non-context-free co-language conjecture for maximally complex words

Let $L_3$ be the language of maximally complex words for nondeterministic automatic complexity over the alphabet ${0,1,2}$, and let $\operatorname{coCFL}$ denote the complements of context-free languages. Non-context-free co-language conjecture.

$$L_3\not\in\operatorname{coCFL}.$$

This is proposed as an analogue of the fact that the set $R=\{x:C(x)\geq\lvert x\rvert\}$ of random words belongs to coRE while $L_3$ is known to be CFL-immune but not coCFL-immune. The conjecture asserts that $L_3$ nevertheless does not belong to coCFL.