  We consider Toda field theories in a classical Euclidean $AdS_2$ background.
We compute the four-point functions of boundary operators in the $a_1$, $a_2$
and $b_2$ Toda field theories. They take the same form as the four-point
functions of generators in the corresponding $\mathcal{W}$-algebras. Therefore
we conjecture that the boundary operators are in one-to-one correspondence with
the generators in the $\mathcal{W}$-algebras.
