  To the first post--Newtonian order, the orbital angular momentum of the
fast--revolving inner binary of the triple system PSR J0337+1715, made of a
millisecond pulsar and a white dwarf, induces an annular gravitomagnetic field
which displaces the line of apsides of the slower orbit of the other, distant
white dwarf by $-1.2$ milliarcseconds per year. The current accuracy in
determining the periastron of the outer orbit is $63.9$ milliarcseconds after
1.38 years of data collection. By hypothesizing a constant rate of measurement
of the pulsar's times of arrivals over the next 10 years, assumed equal to the
present one, it can be argued that the periastron will be finally known to a
$\simeq 0.15$ milliarcseconds level, while its cumulative gravitomagnetic
retrograde shift will be as large as $-12$ milliarcseconds. The competing
post--Newtonian gravitolectric periastron advance due to the inner binary's
masses, nominally amounting to $74.3$ milliarcseconds per year, can be
presently modelled to an accuracy level as good as $\simeq 0.04$
milliarcseconds per year. The mismodelling in the much larger Newtonian
periastron rate due to the quadrupolar term of the multipolar expansion of the
gravitational potential of a massive ring, whose nominal size for PSR
J0337+1715 is $0.17$ degrees per year, might be reduced down to the $\simeq
0.5$ milliarcseconds per year level over the next 10 years. Thus, a first
measurement of such a novel form of gravitomagnetism, although challenging, may
be somehow feasible in a not too distant future.
