By applying the theory of the minimal model program for adjoint foliated structures, we establish the Sarkisov program for algebraically integrable foliations on klt varieties: any two Mori fiber spaces of such structure are connected by a sequence of Sarkisov links. Combining with a result of R. Mascharak, we establish the Sarkisov program for foliations in dimension at most 3 with mild singularities. Log version and adjoint foliated version of the aformentioned Sarkisov programs are also established.
Chen et al. (Wed,) studied this question.