Strong Sarkisov program

Let $X/S \dashrightarrow X'/S'$ be a birational map between Mori fibre spaces, and let

$$X/S \dashrightarrow X_{1}/S_{1} \dashrightarrow X_{2}/S_{2} \dashrightarrow \cdots$$

be the sequence of Sarkisov links constructed by the strong Sarkisov link construction. A birational map between Mori fibre spaces is a square isomorphism if it is induced by an isomorphism of the total spaces over an isomorphism of the bases.

Strong Sarkisov program. The sequence of Sarkisov links constructed in the strong Sarkisov link construction terminates. Equivalently, the birational map $X_{i}/S_{i} \dashrightarrow X'/S'$ becomes a square isomorphism for some $i$.

This is the termination part of the strong Sarkisov program, corresponding to termination of the minimal model program. The supplied context presents it as the remaining step after constructing the successive untwisting links, but gives no resolution status for the conjecture.