Hjorth's reflection argument

Hjorth, assuming $\mathsf{A}\mathsf{D}+\mathsf{Z}\mathsf{F}+\mathsf{D}\mathsf{C}$, showed that there is no sequence of length ${\omega }_2$ consisting of distinct ${\mathrm{\Sigma }}_2^1$-sets. We show that the same theory implies that for $n\ge 0$, there is no sequence of length ${\delta }_{2n+2}^1$ consisting of distinct ${\mathrm{\Sigma }}_{2n+2}^1$ sets. The theorem settles Question 30.21 of Kanamori, which was also conjectured by Kechris.