The birationality conjecture for Landau–Ginzburg moduli of cubic fourfolds and $X_{10}$ varieties

Consider special four-dimensional cubics lying on Noether–Lefschetz loci and special four-dimensional $X_{10}$ varieties, together with their Landau–Ginzburg moduli spaces.

Birationality conjecture. There is an infinite series of such moduli spaces that can be deformed one to another and therefore are birational.

The source relates these deformations to a noncommutative Sarkisov program and suggests that such deformations yield Sarkisov links between the corresponding cubics and $X_{10}$ varieties.