We find a simple relation between a free higher spin partition function on the thermal quotient of ${\mathrm{AdS}}_{d+1}$ and the partition function of the associated d-dimensional conformal higher spin field defined on the thermal quotient of ${\mathrm{AdS}}_{d}$. Starting with a conformal higher spin field defined in ${\mathrm{AdS}}_{d}$, one may also associate to with another conformal field in $d-1$ dimensions, thus iterating AdS/CFT. We observe that in the case of $d=4$, this iteration leads to a trivial 3d higher spin conformal theory with parity-even non-local action: it describes a zero total number of dynamical degrees of freedom and the corresponding partition function is equal to 1.