The Henneberg 2-extension conjecture in three-dimensional rigidity

Let $G=(V,E)$ be a generically 3-isostatic graph, and let $X\subseteq V$ satisfy

$$|X|=5.$$

Let $e$ and $f$ be distinct edges whose vertices lie in $X$. An implied $K_4$ is a complete graph on four vertices formed by implied edges of a graph. The Henneberg 2-extension conjecture. If $G-\{e,f\}$ contains no implied $K_4$ whose vertex set is a subset of $X$, then the Henneberg extension obtained by removing $e$ and $f$ and adjoining a vertex with neighbour set $X$ is generically 3-rigid. This is the open Henneberg Conjecture of Graver, Tay and Whiteley, concerning sufficient conditions for preserving generic rigidity under three-dimensional Henneberg 2-extensions; the corresponding planar theory is well understood through Laman's theorem, but the three-dimensional case remains open.