Contraction conjecture for positively curved spheres

Let $(S^n,g,\operatorname{vol}_g)$ satisfy $\operatorname{Ric}_g\geq\rho g$ with $\rho>0$, and let $g^\rho_{\mathrm{can}}$ be the canonical metric on $S^n$ rescaled so that $\operatorname{Ric}_{g^\rho_{\mathrm{can}}}=\rho g^\rho_{\mathrm{can}}$. A map pushes forward one volume measure onto the other up to a finite constant if its pushforward is proportional to the target measure, and it contracts the corresponding metrics when it is $1$-Lipschitz. Contraction conjecture. There exists a map

$$T:(S^n,g^\rho_{\mathrm{can}},\operatorname{vol}_{g^\rho_{\mathrm{can}}})\longrightarrow(S^n,g,\operatorname{vol}_g)$$

pushing forward $\operatorname{vol}_{g^\rho_{\mathrm{can}}}$ onto $\operatorname{vol}_g$ up to a finite constant and contracting the corresponding metrics. Such a map would imply the spherical spectral comparison conjecture by the contraction principle. The source attributes this as a previously unpublished conjecture and gives no resolution.