  Recent observational surveys of the outer Solar System provide evidence that
Neptune's distant $n$:1 mean-motion resonances may harbor relatively large
reservoirs of trans-Neptunian objects (TNOs). In particular, the discovery of
two securely classified 9:1 resonators, 2015 KE$_{172}$ and 2007 TC$_{434}$, by
the Outer Solar System Origins Survey is consistent with a population of order
$10^4$ such objects in the 9:1 resonance with absolute magnitude $H_r < 8.66$.
This work investigates whether the long-term stability of such populations in
Neptune's $n$:1 resonances can be used to constrain the existence of distant
$5-10M_{\oplus}$ planets orbiting at hundreds of AU. The existence of such a
planet has been proposed to explain a reported clustering in the orbits of
highly eccentric "extreme" trans-Neptunian objects (eTNOs), although this
hypothesis remains controversial. We engage in a focused computational
case-study of the 9:1 resonance, generating synthetic populations and
integrating them for 1 Gyr in the presence of 81 different test planets with
various masses, perihelion distances, eccentricities, and inclinations. While
none of the tested planets are incompatible with the existence of 9:1
resonators, our integrations shed light on the character of the interaction
between such planets and nearby $n$:1 resonances, and we use this knowledge to
construct a simple, heuristic method for determining whether or not a given
planet could destabilize a given resonant population. We apply this method to
the currently estimated properties of Planet 9, and find that a large
primordial population in the 15:1 resonance (or beyond), if discovered in the
future, could potentially constrain the existence of this planet.
