Degree-sum conjecture for graph rigidity

For a graph $G$, define

$$\eta(G)=\min_{uv\notin E(G)}\bigl(\deg_G(u)+\deg_G(v)\bigr),$$

and let $g(n,d)$ be the smallest integer such that every $n$-vertex graph $G$ with $\eta(G)\geq g(n,d)$ is $d$-rigid. Degree-sum conjecture. If $n,d\in\mathbb{N}$ and $n>2d+2$, then

$$g(n,d)=n+d-2.$$

This is proposed after noting that the minimum-degree lower bound gives $g(n,d)\geq n+d-2$ in this regime; the paper does not state a resolution of the proposed equality.