Triviality conjecture for Legendrian unknot racks

Let $L$ be a Legendrian unknot, let $tb(L)$ denote its Thurston–Bennequin number, and let $n$-Legendrian racks be the racks associated to $L$ by the $n$-Legendrian rack construction. Triviality conjecture. If

$$\mid tb(L) \mid = 2^k$$

for some $k\in\mathbb{N}$, then every $n$-Legendrian rack is trivial. This conjecture was motivated by automated theorem-prover experiments for several cusped Legendrian unknots; a general proof was not obtained, so its status remains open.