Analytic torsion formula for cones over odd-dimensional spheres

Let $C_\alpha S^{2p-1}_{l\sin\alpha}$ be the cone of angle $\alpha$ and length $l>0$ over the odd-dimensional sphere $S^{2p-1}$, equipped with the standard metric induced by the immersion in $\mathbb{R}^{m+1}$ and absolute boundary conditions. Assume $p>0$. Analytic torsion conjecture. The analytic torsion satisfies

$$\begin{aligned} \log T(C_\alpha S^{2p-1}_{l\sin\alpha})={}&\frac{1}{2}\log \operatorname{Vol}(C_\alpha S^{2p-1}_{l\sin\alpha})\\ &+\sum_{j=0}^{p-1}\frac{2^{p-j}}{j!(2(p-j)-1)!!}\sum_{h=0}^{j}\binom{j}{h}\frac{(-1)^h\operatorname{csc}^{2(j-h)}\alpha}{2(p-j+h)-1}\frac{(2p-1)!\sin^{2p-1}\alpha}{4^p(p-1)!}. \end{aligned}$$

This formula agrees with the corresponding expression for cones over the circle and over the $3$-sphere, where it is a theorem for $p<3$; the conjecture concerns the additional contribution of the singularity in higher odd dimensions.