Finite-index closure failure for the submonoid membership problem

Let $G$ be a group and let $H\leq G$ be a subgroup of index $2$. Write $\operatorname{SMON}(K)$ for the submonoid membership problem in a group $K$. Finite-index submonoid membership conjecture. There is a group $G$ with a subgroup $H$ of index $2$ such that $\operatorname{SMON}(H)$ is decidable but $\operatorname{SMON}(G)$ is undecidable. This conjecture asks whether decidability of the submonoid membership problem is preserved when passing between a group and a finite-index subgroup; the paper presents evidence for a negative answer, while the existence of such an example remains open.