Note that we use a slightly different definition for $Q_n(\xi )$ compared to [26]. In particular, $Q_n(\xi )$ defined here is equal to $Q_{n+1}(\xi )$ in [26].

The similar construction is also valid for more holographic backgrounds such as [36, 37, 38, 39, 40, 41, 42, 43].

We have used the AdS/CFT dictionary, $R^4/{\ell }_s^4=\lambda$, where $R$ is the AdS curvature and ${\ell }_s$ is the string length, to express the large AdS curvature expansion in terms of the large-$\lambda$ expansion.

We have independently verified this conclusion using the residue theorem. The convergence condition for the contour integral depends on the sign of $\ln |z{|}^2$. Assuming $\mathrm{Re}(S)>0$ requires closing the contour in the left half-plane, which selects the region $|z|<1$. Conversely, assuming $\mathrm{Re}(S)<0$ dictates closing the contour to the right, corresponding to $|z|>1$. While these domains are complementary, the sign reversal from the change in contour orientation cancels the sign difference between the radial integrals using the formal integral relation ${\int }_0^{1}dr{r}^k=-{\int }_1^{\infty }dr{r}^k$. Consequently, the final result is independent of the assumption regarding $\mathrm{Re}(S)$.

Note we should remove the supergravity amplitude $1/(STU)$ for focusing the stringy effects.

We note our result for $c_{1,1}$ disagrees with that in the Ref. [23], corresponding to the ${\lambda }^{-3/2}$ term in Eq. (4.6) of that reference. Converting to our basis, their expression implies $c_{1,1}=48{\zeta }_2-240{\zeta }_3$. We find that this discrepancy arises from performing the $s$-integration prior to the $\delta$-summation in [23], which omits a certain subtle contribution (more precisely, a term with a $0/0$ structure). We have explicitly verified that including such contribution restores precise agreement. We thank Alexander Zhiboedov for very helpful discussions on this point.