Unique gear-mapping parameter for prescribed prevertices

Let $t_1,t_2$ satisfy

$$0<t_1<t_2<\pi/2.$$

For a real parameter $\lambda$, let $f$ be the solution of

$${\mathcal{S}}_f=R_{t_1,t_2,\lambda}$$

normalized by $J_f(0)=(0,1,0)$, and call $f$ a gear mapping when its image is a gear-shaped domain. Uniqueness conjecture. For every such pair $t_1,t_2$, there is a unique $\lambda\in\mathbb{R}$ for which $f$ is a gear mapping. This conjecture is motivated by numerical examples of repositioning the gear center; the supplied text gives no proof or resolution, so the uniqueness remains open.