Graver's maximality conjecture in dimension three

Let ${\cal R}_{3,n}$ be the generic $3$-dimensional rigidity matroid on $E(K_n)$, and let an abstract $3$-rigidity matroid be an abstract $d$-rigidity matroid with $d=3$. Graver's maximality conjecture. The generic $3$-dimensional rigidity matroid is the unique maximal abstract $3$-rigidity matroid on $E(K_n)$. The conjecture is a long-standing open problem for $d=3$; the corresponding general conjecture is false for $d\geq4$.