  With the growth of comparative exoplanetology, it is increasingly clear that
investigating the relationships between inner and outer planets plays a key
role in discriminating between competing formation and evolution models. To do
so, it is important to probe the inner region of systems hosting long-period
giants in search for undetected lower-mass planetary companions. In this work
we present our results on the K-dwarf star BD-11~4672, already known to host a
long-period giant planet, as the first output of a subsample of the GAPS
programme specifically aimed at assessing the impact of inefficient migration
of planets formed beyond the snowline by searching for Neptune-mass and
super-Earths planetary companions of known longer-period giants. The
high-precision HARPS-N observations of BD-11~4672 are used in conjunction with
literature time series in order to search for additional inner planetary
signals to be fitted using differential evolution Markov chain Monte Carlo. The
long-term stability of the new orbital solutions is tested by N-body dynamical
simulations. We report the detection of BD-11~4672~c, a new Neptune-mass planet
with an orbital period of $74.20_{-0.08}^{+0.06}\,\rm{d}$, eccentricity
$0.40_{-0.15}^{+0.13}$, semimajor axis $0.30\pm0.01\,\rm{au}$ and minimum mass
$15.37_{-2.81}^{+2.97}\,M_\oplus$ orbiting slightly outside the inner edge of
the optimistic circumstellar habitable zone. In order to assess its impact on
the dynamical stability of the habitable zone we compute the angular momentum
deficit of the system, showing that planet c has a severe negative impact on
the stability of possible additional lower-mass temperate planets. The
BD-11~4672 system is notable for its architecture, hosting both a long-period
giant planet and an inner lower-mass planet, the latter being also among the
most eccentric Neptune-mass planets known at similar periods.
