  Aims: It has recently been shown that the fast migration of Titan could be
responsible for the current 26.7{\deg}-tilt of Saturn's spin axis. We aim to
quantify the level of generality of this result and to measure the most likely
sets of parameters. We also aim to determine the obliquity that Saturn will
reach in the future.
  Methods: We explore a broad range of parameters and propagate numerically the
orientation of Saturn's spin axis both backward and forward in time.
  Results: In the adiabatic regime, the likelihood of reproducing Saturn's
current spin-axis orientation is maximised for primordial obliquities between
about 2{\deg} and 7{\deg}. For a slightly faster migration than expected from
radio-science experiments, non-adiabatic effects even allow for exactly null
primordial obliquities. Starting from such small tilts, Saturn's spin axis can
evolve up to its current state provided that: i) the semi-major axis of Titan
changed by more than 5% of its current value, and ii) its migration rate does
not exceed 10 times the nominal measured rate. In comparison, observational
data suggest that the increase of Titan's semi-major axis exceeded 50% over 4
Gyrs, and error bars imply that the current migration rate is unlikely to be
larger than 1.5 times its nominal value.
  Conclusions: If Titan did migrate substantially before today, tilting Saturn
from a small obliquity is not only possible, but it is the most likely
scenario. Saturn's obliquity is expected to be still increasing today and could
exceed 65{\deg} in the future. Maximising the likelihood would also put strict
constraints on Saturn's polar moment of inertia. However, the possibility
remains that Saturn's primordial obliquity was already large, for instance as a
result of a massive collision. The unambiguous distinction between these two
scenarios would be given by a precise measure of Saturn's polar moment of
inertia.
