Integrability of the Hirota–Kimura discretization of the general Clebsch flow

Let the Clebsch system be discretized by the map $f$ defined by the displayed Hirota–Kimura discretization, and let an HK-basis be a basis of quadratic monomials whose associated null-space determines a pencil of quadrics containing the map's orbits. For the general flow of the Clebsch system, use

$$\Phi_0=(p_1^2,p_2^2,p_3^2,m_1^2,m_2^2,m_3^2,1).$$

Clebsch discretization conjecture. All claims of the relevant integrability theorems for the special Clebsch discretizations also hold for this discretization of the general Clebsch flow, with the HK-basis $\Phi_0$ as given above.

This asserts that the general-flow discretization has the same HK-basis and integrability properties as the previously studied Clebsch discretizations. The supplied text gives no resolution status beyond presenting the claim in a conjecture environment.