Kálmán's uniqueness conjecture for braid-positive Legendrian links

Let $\beta$ be a positive braid, let $L_\beta$ be its braid-positive link closure, and let the corresponding Legendrian closure be represented by the front diagram $f_\beta$. A Legendrian link is braid-positive if its underlying link type is the closure of a positive braid. Kálmán's conjecture. Any braid-positive Legendrian link is a stabilization of the corresponding Legendrian closure shown in the paper's figure. In particular, braid-positive links are Legendrian simple. This conjecture asserts that the canonical Legendrian closure is essentially the only Legendrian representative of a braid-positive link type, up to stabilization; the supplied source does not state a resolution.