Scaling conjecture for Legendrian unknot racks

Let $L_1$ and $L_2$ be Legendrian unknots, let $tb(L_i)$ denote their Thurston–Bennequin numbers, and let $\mathcal{LR}(L_i)$ denote their Legendrian racks. Scaling conjecture. If

$$tb(L_1)=2^k tb(L_2)$$

for some $k\in\mathbb{Z}_{\geq 0}$, then

$$\mathcal{LR}(L_1)=\mathcal{LR}(L_2).$$

This conjecture was postulated from experimental data for several cusped Legendrian unknots and from the proof strategy for another theorem. Since automated theorem provers did not readily check rack isomorphisms for nontrivial racks, no general experimental verification was available, and the conjecture remains open.