Let
π
⊗
σ
\pi \otimes \sigma
be a supercuspidal representation of
G
L
(
2
n
)
×
S
O
(
2
n
)
\mathrm {GL}(2n) \times \mathrm {SO}(2n)
over a
p
p
-adic field with
π
\pi
selfdual, where
S
O
(
2
n
)
\mathrm {SO}(2n)
stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to
S
O
(
6
n
)
\mathrm {SO}(6n)
, Goldberg and Shahidi defined a pairing
R
R
between the matrix coefficients of
π
\pi
and
σ
\sigma
which controls the residue of the standard intertwining operator. The elliptic part
R
ell
R_\text {ell}
of
R
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
n
, under the assumption that
π
\pi
does not come from
S
O
(
2
n
+
1
)
\mathrm {SO}(2n+1)
.