Cuspless sub-Riemannian exponential-map conjecture in $SE(3)$

Let $\widetilde{Exp}_e$ be the reduced exponential map, let $\mathcal{D}$ be its domain, let $\mathcal{R}$ be its range, and let $\mathcal{C}$ and $s_{\max}$ be as defined in the paper. For a set $S$, write $\operatorname{int}(S)$ for its interior. Cuspless sub-Riemannian exponential-map conjecture. The map

$$\widetilde{Exp}_e:\mathcal{D}\to\mathcal{R}$$

is a homeomorphism for the subspace topologies, and its restriction

$$\widetilde{Exp}_e:\operatorname{int}(\mathcal{D})\to\operatorname{int}(\mathcal{R})$$

is a diffeomorphism. Moreover,

$$\partial\mathcal{R}=S_B\cup S_R\cup S_L,$$

where

$$S_B=\{\widetilde{Exp}_e(\boldsymbol{\lambda}(0),s_{\max}(\boldsymbol{\lambda}(0)))\mid \boldsymbol{\lambda}(0)\in\mathcal{C}\},$$ $$S_R=\{\widetilde{Exp}_e(\boldsymbol{\lambda}(0),s)\mid \boldsymbol{\lambda}(0)\in\mathcal{C},\ \lambda_4(0)^2+\lambda_5(0)^2=1,\ s>0\},$$

and

$$S_L=\{(\mathbf{0},R)\in SE(3)\mid R\mathbf{e}_z\cdot\mathbf{e}_z\geq 0\}.$$

The conjecture is motivated by numerical experiments and by the corresponding result for the two-dimensional setup; its validity in this three-dimensional position-and-direction space remains open.