The Elliptic Painlevé Lax Equation vs. van Diejen's $8$-Coupling Elliptic Hamiltonian

The $8$-parameter elliptic Sakai difference Painlevé equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrödinger equation for the $BC_1$ $8$-parameter ‘relativistic’ Calogero–Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlevé–Calogero correspondence to the highest level in the two hierarchies.