Uniform winning strategies for the synchronization games on subclasses of finite automata

The pseudovariety $\mathbf{DS}$ consists of all finite monoids whose regular $D$-classes form subsemigroups. We exhibit a uniform winning strategy for Synchronizer in the synchronization game on every synchronizing automaton whose transition monoid lies in $\mathbf{DS}$, and we prove that $\mathbf{DS}$ is the largest pseudovariety with this property.