  Context. The paths followed by the known extreme trans-Neptunian objects
(ETNOs) effectively avoid direct gravitational perturbations from the four
giant planets, yet their orbital eccentricities are in the range between
0.69-0.97. Solar system dynamics studies show that such high values of the
eccentricity can be produced via close encounters or secular perturbations. In
both cases, the presence of yet-to-be-discovered trans-Plutonian planets is
required.
  Aims. If the high eccentricities of the known ETNOs are the result of
relatively recent close encounters with putative planets, the mutual nodal
distances of sizeable groups of ETNOs with their assumed perturber may still be
small enough to be identifiable geometrically. In order to confirm or reject
this possibility, we used Monte Carlo random search techniques.
  Methods. Two arbitrary orbits may lead to close encounters when their mutual
nodal distance is sufficiently small. We generated billions of random planetary
orbits with parameters within the relevant ranges and computed the mutual nodal
distances with a set of randomly generated orbits with parameters consistent
with those of the known ETNOs and their uncertainties. We monitored which
planetary orbits had the maximum number of potential close encounters with
synthetic ETNOs and we studied the resulting distributions.
  Results. We provide narrow ranges for the orbital parameters of putative
planets that may have experienced orbit-changing encounters with known ETNOs.
  Conclusions. Our calculations suggest that more than one perturber is
required if scattering is the main source of orbital modification for the known
ETNOs. Perturbers might not be located farther than 600 AU and they have to
follow moderately eccentric and inclined orbits to be capable of experiencing
close encounters with multiple known ETNOs.
