Handlebody subgroups in a mapping class group

Suppose subgroups $A,B<MCG(S)$ in the mapping class group of a closed orientable surface $S$ are given and let $⟨A,B⟩$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A, B \rangle \simeq A*_{A \cap B} B$forhandlebodysubgroups$A,B$.