  Toda Conformal Field Theories (CFTs) form a family of two-dimensional CFTs
indexed by semisimple and complex Lie algebras. One of their remarkable
features is that they are natural generalizations of Liouville CFT that enjoy
an enhanced level of symmetry, prescribed by $W$-algebras. They likewise admit
a probabilistic formulation in terms of Gaussian Multiplicative Chaos.
  Based on this probabilistic framework, this second article in a two-part
series is dedicated to providing a first step towards integrability of these
theories. In this perspective we prove the Fateev-Litvinov formula for a family
of three-point correlation functions associated to the $\mathfrak{sl}_3$ Toda
CFT. This result is the analog of the celebrated DOZZ formula in Liouville CFT.
Our method of proof features techniques inspired by the physics literature
together with probabilistic ones that naturally arise within the setting of
Toda theories.
