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Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits?

An example is given by the irrational ellipsoid in $(\mathbb{R}^4,\omega_\text{st})$, which is not overtwisted. The only other example that I know of are quotients of the last example on lens spaces. Actually, contact manifolds admitting only a finite number of periodic orbit are proven to "rarely exist" (https://arxiv.org/pdf/0809.5088.pdf).

(I am aware that under non-degeneracy condition on the contact form, the authors in https://arxiv.org/pdf/1701.02262.pdf prove that there are $2$ or $\infty$-many periodic Reeb orbits, but as far as I know nothing more is known if the contact structure is overtwisted).

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    $\begingroup$ It is an open question for a 3-manifold: The conjecture is that a contact 3-manifold either has two (if it's a quotient of 3-sphere) or infinitely many embedded Reeb orbits. $\endgroup$
    – Chris Gerig
    Commented Jul 19, 2019 at 15:49
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    $\begingroup$ And the introduction of Gardiner-Hutchings-Pomerleano's paper that you reference (Corollary 1.6 and footnote 3) shows that, assuming nondegeneracy, there are infinitely many on overtwisted Lens spaces (including 3-sphere). $\endgroup$
    – Chris Gerig
    Commented Jul 19, 2019 at 16:19

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