  Let {\sigma}\otimes{\pi} be a supercuspidal representation of SO(2n) \times
GL(2n) over a p-adic field with {\pi} selfdual, where SO(2n) stands for a
quasisplit even special orthogonal group. In order to study its normalized
parabolic induction to SO(6n), Goldberg and Shahidi defined a pairing R between
the matrix coefficients of {\sigma} and {\pi} which controls the residue of the
standard intertwining operator. The elliptic part of R is conjectured to be
related to twisted endoscopic transfer. Based on Arthur's endoscopic
classification and Spallone's improvement of Goldberg-Shahidi program, we will
verify some of their predictions for general n, under the assumption that {\pi}
does not come from SO(2n+1).
