It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an overtwisted contact structure has closed orbits, a theorem due to Hofer. I wonder what is currently know about Reeb orbits of overtwisted contact structures on manifolds that are not closed. Is there any known example of an open overtwisted $3$-manifold whithout any closed Reeb orbits?
I've been looking into $\mathbb{R}^3$ with its canonical overtwisted contact form for a possible example, without much success. Using cylindrical coordinates $(r, \theta, z)$ for $\mathbb{R}^3$, this contact form is given by $$\lambda_{OT} = \cos(r)dz + r\sin(r)d\theta.$$ Since I couldn't find any explicit calculation of the Reeb vector field associated to this form in any book or paper, I tried to determine it on my on and arrived at $$ R(r, \theta, z)= \begin{cases} 2\left(0, \frac{\sin(r)}{\sin(2r) + 2r}, \frac{\sin(r)+ r\cos(r)}{\sin(2r) + 2r}\right), \text{ if } r \neq 0, \\ \left(0, \frac{1}{2}, 1 \right) \text{ if } r = 0, \end{cases} $$ which is continuous at $r = 0$ and satisfies the equalities $\lambda_{OT}(R) = 1$ and $d\lambda_{OT}(R, \cdot) = 0$. I don't know if this is completely correct because I had a feeling the Reeb vector field at the line $r=0$ should be $\partial_z$, since the contact form reduces to $dz$ in this case. Anyway, I was unable to give explicit integral curves for the vector field $R$ or to conclude if there are any closed orbits, so I'm still not sure whether $(\mathbb{R}^3, \lambda_{OT})$ is an example of overtwisted $3$-manifold without closed Reeb orbits or not.