An Elliptic Macdonald-Morris Conjecture and Multiple Modular Hypergeometric Sums

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Inversions of integral operators and elliptic beta integrals on root systems (2004, methodology): “For n an integer the elliptic shifted factorial is defined by [28] (a; q, p)n = Γ(aq; p, q) Γ(a; p, q) (this was denoted as θ(a; p; q)n in [6, 7, 24]).”
Elliptic hypergeometric series on root systems (2002, methodology): “tivariable elliptic analogue of Jackson’s 8W7 summation formula (cf. (7.1) below) and conjectured another one. We will refer to these as Warnaar’s ﬁrst and second summation. Van Diejen and Spiridonov [DS1] found that the second summation may be derived from a certain conjectured elliptic Selberg int”
Transformations of elliptic hypergometric integrals (2003, background): “For m = 0, we obtain the following integral conjectured by Spiridonov [32]: Corollary 4.2.”
Transformations of elliptic hypergeometric integrals (2010, background): “A direct proof of the corollary has since been given by Spiridonov [33]; see also the remark following the second proof of Theorem 6.1 below.”