Let
A
A
be a principally polarized abelian variety of dimension
g
g
over a number field
K
K
. Assume that the image of the adelic Galois representation of
A
A
is an open subgroup of
G
S
p
2
g
(
Z
^
)
GSp_{2g}(\hat {\mathbb {Z}})
. Then there exists a positive integer
m
m
so that the Galois image of
A
A
is the full preimage of its reduction modulo
m
m
. The least
m
m
with this property, denoted
m
A
m_A
, is called the image conductor of
A
A
. Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for
m
A
m_A
, in terms of standard invariants of
A
A
, in the case that
A
A
is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.