DLS generation conjecture for outer automorphisms of special groups

Let $G$ be a special group. A DLS automorphism is one of the automorphisms appearing in Theorems~ and~; those appearing in Theorem~ are the relevant ones for $\operatorname{Out}_{\rm cmp}(G)$. DLS generation conjecture. The DLS automorphisms appearing in Theorem~ generate a finite-index subgroup of $\operatorname{Out}_{\rm cmp}(G)$, and finitely many such automorphisms suffice. The DLS automorphisms appearing in Theorems~ and~ generate a finite-index subgroup of $\operatorname{Out}(G)$, and finitely many such automorphisms suffice. In particular, $\operatorname{Out}(G)$ and $\operatorname{Out}_{\rm cmp}(G)$ are finitely generated. This conjecture would give finite-generation and finite-index generation results for the outer automorphism groups of special groups; the supplied text says that the preceding theorems provide significant evidence, but does not resolve the conjecture.