  We know little about the outermost exoplanets in planetary systems, because
our detection methods are insensitive to moderate-mass planets on wide orbits.
However, debris discs can probe the outer-planet population, because dynamical
modelling of observed discs can reveal properties of perturbing planets. We use
four sculpting and stirring arguments to infer planet properties in 178
debris-disc systems from the ISPY, LEECH and LIStEN planet-hunting surveys.
Similar analyses are often conducted for individual discs, but we consider a
large sample in a consistent manner. We aim to predict the population of
wide-separation planets, gain insight into the formation and evolution
histories of planetary systems, and determine the feasibility of detecting
these planets in the near future. We show that a `typical' cold debris disc
likely requires a Neptune- to Saturn-mass planet at 10-100 au, with some
needing Jupiter-mass perturbers. Our predicted planets are currently
undetectable, but modest detection-limit improvements (e.g. from JWST) should
reveal many such perturbers. We find that planets thought to be perturbing
debris discs at late times are similar to those inferred to be forming in
protoplanetary discs, so these could be the same population if newly formed
planets do not migrate as far as currently thought. Alternatively, young
planets could rapidly sculpt debris before migrating inwards, meaning that the
responsible planets are more massive (and located further inwards) than
debris-disc studies assume. We combine self-stirring and size-distribution
modelling to show that many debris discs cannot be self-stirred without having
unreasonably high masses; planet- or companion-stirring may therefore be the
dominant mechanism in many (perhaps all) debris discs. Finally, we provide
catalogues of planet predictions, and identify promising targets for future
planet searches.
