Conjecture on the asymptotic landing points of constriction curves

Let $\beta_{\ell,k}=(0,s_{\ell,k})$ be the points on the $s$-axis, where $s_{\ell,1},s_{\ell,2},\dots$ are the positive zeros of the Bessel function $J_\ell$, and let $\mathcal C_{\ell,k}$ denote the constriction curve accumulating to $\beta_{\ell,k}$. Landing-point conjecture. Each constriction curve lands at some $\beta_{\ell,k}$, i.e., coincides with some $\mathcal C_{\ell,k}$. The preceding results identify the points $\beta_{\ell,k}$ as the limit set of the constriction submanifold and show that exactly one known constriction curve accumulates regularly to each such point; the conjecture asserts that every constriction curve is among these curves.