The EDT0L nonmembership conjecture for rank-one free inverse monoid word problems

Let $\operatorname{FIM}_1$ be the free inverse monoid of rank $1$. Write $\operatorname{WP}(\operatorname{FIM}_1)$ and $\iota(\operatorname{FIM}_1)$ for its two word problem languages, and let EDT0L denote the class of languages generated by deterministic extended tabled 0L systems. EDT0L nonmembership conjecture. Neither $\operatorname{WP}(\operatorname{FIM}_1)$ nor $\iota(\operatorname{FIM}_1)$ is EDT0L. These languages are nevertheless known to be ET0L, so the conjecture would separate the ET0L and EDT0L descriptions for these rank-one word problems.