3 problems

Equivalence conjecture. Every two maximal complete $1$-systems of loops on $F$ are equivalent.

Nonexistence conjecture. There does not exist a maximum complete 1-system $\Gamma$ whose dual cube complex $C(\Gamma)$ is $2$-dimensional.

Factorial growth conjecture. The number of $\operatorname{Mod}^*(S)$-inequivalent complete 1-systems of size $2g+1$ grows at least factorially in $g$.