  We analytically work out the orbital effects induced by the gravitomagnetic
spin-octupole moment of an extended spheroidal rotating body endowed with
angular momentum $\boldsymbol{S}$ and quadrupole mass moment $J_2$. Our
results, proportional to $S J_2 c^{-2}$, hold for an arbitrary orientation of
the body's symmetry axis $\boldsymbol{\hat{S}}$ and a generic orbital
configuration of the test particle. Such effects may be measurable, in
principle, with a dedicated spacecraft-based mission to Jupiter. For a
moderately eccentric and fast path, the gravitomagnetic precessions of the node
and the pericenter of a dedicated orbiter could be as large as
$400~\textrm{milliarcseconds~per~year}$ or even
$1,600-4,000~\textrm{milliarcseconds~per~year}$ depending on the orientation of
its orbital plane in space. Numerical simulations of the Earth-probe range-rate
signal confirm such expectations since its magnitude reaches the $\simeq
0.03-0.3~\textrm{millimeter~per~second}$ level after just 1 day. The precision
of the current two-way Ka-band Doppler measurements of the spacecraft Juno,
presently orbiting Jupiter, amounts to $\simeq
0.003~\textrm{millimeter~per~second}$ after $1,000$ seconds. Also other general
relativistic effects could be measurable, including also those proportional to
$GMJ_2c^{-2}$, never put to the test so far. Most of the competing Newtonian
signals due to the classical multipoles of the planet's gravity field have
quite different temporal signatures with respect to the post-Newtonian ones,
making, thus, potentially easier disentangling them.
