  We consider the conformal $A_{n}$ Toda theory in AdS$_{2}$. Due to the bulk
full Virasoro symmetry, this system provides an instance of a non-gravitational
$\text{AdS}_{2}$/$\text{CFT}_{1}$ correspondence where the 1d boundary theory
enjoys enhanced "$\frac{1}{2}$-Virasoro" symmetry. General boundary correlators
are expected to be captured by the restriction of chiral correlators in a
suitable $\mathcal WA_{n}$ Virasoro extension. At next-to-leading order in weak
coupling expansion they have been conjectured to match the subleading terms in
the large central charge expansion of the dual $\mathcal WA_{n}$ correlators.
We explicitly test this conjecture on the boundary four point functions of the
Toda scalar fields dual to $\mathcal WA_{n}$ generators with next-to-minimal
spin 3 and 4. Our analysis is valid in the generic rank case and extends
previous results for specific rank-2 Toda theories. On the AdS side, the
extension is straightforward and requires the computation of a finite set of
tree Witten diagrams. This is due to simple rank dependence and selection rules
of cubic and quartic couplings. On the boundary, the CFT calculation is made
feasible by exploiting crossing symmetry and specific meromorphic properties of
the $\mathcal W$-algebra correlators at large central charge. We present the
required 4-point functions in closed form for any rank and verify the
bulk-boundary correspondence in full details.
