Isospectral spherical space forms and orbifolds of highest volume

We prove that $\mathrm{vol}(S^d)\mathrm{/}8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $d\ge 5$. Furthermore, we show that $\mathrm{vol}(S^{2n-1})\mathrm{/}11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $n\ge 11$ and $n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$, or $n\ge 7$ and $n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}5)$, or $n\ge 3$ and $n\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}5)$.