A proof of a multivariable elliptic summation formula conjectured by Warnaar

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We prove a multivariable elliptic analogue of Jackson's 8W7 summation formula, which was recently conjectured by S.O.Warnaar.

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Transformations of elliptic hypergometric integrals (2003, background): “The fact that this sum simplifies underlies Rosengren’s arguments in section 7 of [25].”
Advanced Determinant Calculus: A Complement (2005, background): “For example, determinant evaluations involving elliptic functions have come into the picture in the theory of multiple elliptic hypergeometric series, see [88, 134, 138, 139, 141, 163, 174].”
Essays on the theory of elliptic hypergeometric functions (2008, background): “It follows from the Cnelliptic beta integral of the type II [32] and was proven for the first time recursively in the paper [45].”
$BC_n$-symmetric abelian functions (2004, methodology): “2] (also proved in [18]), which in our notation reads”
Transformations of elliptic hypergeometric integrals (2010, background): “The fact that this sum simplifies underlies Rosengren’s arguments in Section 7 of [26].”