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

The pseudovariety $\mathbf{D}\mathbf{S}$ 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 $\mathbf{D}\mathbf{S}$, and we prove that $\mathbf{D}\mathbf{S}$ is the largest pseudovariety with this property.