Landau–Ginzburg families conjecture for Sarkisov links

A Sarkisov link is a birational map, or birational cobordism, connecting two Mori fibrations. Landau–Ginzburg families conjecture. One-parameter families of Landau–Ginzburg models parameterize Sarkisov links. The conjecture proposes a connection between moduli of Landau–Ginzburg models, circuits and birational geometry; it is treated in DKK, and the source gives no resolution.