Odd-dimensional lens-space Yamabe counterexamples

Let $n\ge 5$ be odd and let $L_{3q}^n=S^n\mathrm{/}{C}_{3q}$ be the standard lens space, where $q>3$ is prime. We prove that for all sufficiently large $q$, $$\sigma (L_{3q}^n)>(3q{)}^{-2\mathrm{/}n}Y(S^n)=Y(L_{3q}^n,[g_{\mathrm{r}\mathrm{d}}])\mathrm{.}$$ Here $[g_{\mathrm{r}\mathrm{d}}]$ denotes the quotient-round conformal class. Combining spin bordism with the surgery theorem of Ammann–Dahl–Humbert, we establish $$\sigma (L_{3q}^n)\ge \min \{q^{-2\mathrm{/}n}Y(S^n),{\mathrm{\Lambda }}_n\},$$ where ${\mathrm{\Lambda }}_n>0$ depends only on $n$. For sufficiently large $q$, this exceeds the quotient-round value, showing that the quotient-round conformal class need not maximize the smooth Yamabe invariant in any odd dimension $n\ge 5$.