  Exomoons are a missing piece of exoplanetary science. Recently, two promising
candidates were proposed, Kepler-1625 b-I and Kepler-1708 b-I. While the latter
still lacks a dynamical analysis of its stability, Kepler-1625 b-I has already
been the subject of several studies regarding its stability and origin.
Moreover, previous works have shown that this satellite system could harbour at
least two stable massive moons. Motivated by these results, we explored the
stability of co-orbital exomoons using the candidates Kepler-1625 b-I and
Kepler-1708 b-I as case studies. To do so, we performed numerical simulations
of systems composed of the star, planet, and the co-orbital pair formed by the
proposed candidates and another massive body. For the additional satellite, we
varied its mass and size from a Mars-like to the case where both satellites
have the same physical characteristics. We investigated the co-orbital region
around the Lagrangian equilibrium point $L_4$ of the system, setting the
orbital separation between the satellites from $\theta_{min} = 30^{\circ}$ to
$\theta_{max} = 90^{\circ}$. Our results show that stability islands are
possible in the co-orbital region of Kepler-1708 b-I as a function of the
co-orbital companion's mass and angular separation. Also, we identified that
resonances of librational frequencies, especially the 2:1 resonance, can
constrain the mass of the co-orbital companion. On the other hand, we found
that the proximity between the host planet and the star makes the co-orbital
region around Kepler-1625 b-I unstable for a massive companion. Finally, we
provide TTV profiles for a planet orbited by co-orbital exomoons.
