Integrability conjecture for Hirota–Kimura discretizations of quadratic systems

Let a dynamical system be algebraically completely integrable and have a quadratic vector field. Consider its Hirota–Kimura discretization.

Hirota–Kimura integrability conjecture. The Hirota–Kimura discretization remains algebraically completely integrable.

The conjecture proposes that algebraic complete integrability survives this discretization for every system in the stated class. The paper presents it as a hypothesis motivated by numerical experiments and notes that a general structural explanation and systematic non-computational proofs remain unknown.