Establish whether the growth sensitivity of the ShortLex language holds

Establish whether the ShortLex language SL(FIM_X) of a free inverse monoid is growth sensitive, namely, whether forbidding every subword f in Sub(SL(FIM_X)) strictly decreases its exponential growth rate.

Background

The growth-tightness argument would follow naturally if the ShortLex language SL(FIM_X) were growth sensitive: removing words containing any forbidden subword would then strictly reduce its exponential growth rate. The paper invokes a general theorem for languages generated by irreducible unambiguous context-free grammars, but the grammar for SL(FIM_X) is not irreducible. The authors therefore state that they are unable to prove the required growth-sensitivity property directly, leaving whether it holds as an unresolved issue.

References

The basic idea is to consider $L=SL(FIM_X)$ and prove that this language is growth sensitive. (However, we will not quite be able to prove this.)

— Green languages and growth of free inverse monoids  (2609.35251 - Bodart et al., 28 Sep 2026) in Section 10, subsection Proof, paragraph beginning “The basic idea is to consider”