Completeness conjecture for Legendrian graph stabilizations and vertex twists

Let $G$ be a graph and let $g:G\rightarrow S^3$ be a smooth embedding. A Legendrian realization of $g$ is a Legendrian embedding of $G$ in $(S^3,\xi_{std})$ representing the smooth embedding $g$. Legendrian graph equivalence conjecture. Any two Legendrian realizations $g_1$ and $g_2$ of $g$ are related, up to Legendrian isotopy, by a sequence of edge stabilizations, vertex stabilizations, and vertex twists.

Edge stabilization, vertex stabilization, and vertex twist are local operations on Legendrian graphs that preserve their smooth isotopy class. The conjecture proposes that these operations completely classify Legendrian realizations within a fixed smooth isotopy class; the preceding discussion explains that edge stabilizations alone are insufficient, while no resolution is given here.