  We refine the statement of the denominator and evaluation conjectures for
affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and prove the
first non-trivial cases of these conjectures. Our results provide a
q-deformation of the computation of genus 1 conformal blocks via elliptic
Selberg integrals by Felder-Stevens-Varchenko. They allow us to give precise
formulations for the affine Macdonald conjectures in the general case which are
consistent with computer computations.
  Our method applies recent work of the second named author to relate these
conjectures in the case of $U_q(\widehat{\mathfrak{sl}}_2)$ to evaluations of
certain theta hypergeometric integrals defined by Felder-Varchenko. We then
evaluate the resulting integrals, which may be of independent interest, by
well-chosen applications of the elliptic beta integral introduced by
Spiridonov.
