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The boundary of the neutron $i_{13/2}$ single-particle energy is investigatedwith exact shell-model calculations, where random two-body interactions areadopted to overcome the bias from effective interactions. Excitation energiesof $3^-_1$ state in $^{134}$Te and $^{136}$Xe, as well as those of $13/2^+_1$states in $^{135}$Te and $^{137}$Xe, are taken as touchstones of our samplings.A robust upper limit of $\varepsilon_{i13/2}<3.1$ MeV emerges in ourcalculations, given the mixing of $i_{13/2}$ single-neutron configuration and$f_{7/2}\otimes 3^-$ configuration in $13/2^+_1$ states of $N=83$ isotones.

Robust upper limit on the neutron single-particle energy of the $i_{13/2}$ orbit