The Elliptic Painlev\'e Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian

The 8-parameter elliptic Sakai difference Painlev'e equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schr"odinger equation for the $B{C}_1$ 8-parameter ‘relativistic’ Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlev'e-Calogero correspondence to the highest level in the two hierarchies.