Krivelevich–Lew–Michaeli edge-counting conjecture for graph rigidity

Let $f(n,d)$ be the smallest integer such that every $n$-vertex graph with minimum degree at least $f(n,d)$ is $d$-rigid. A graph is $d$-rigid when every generic framework in $\mathbb{R}^d$ is rigid. Krivelevich–Lew–Michaeli's conjecture. For $1\leq d<n$,

$$f(n,d)\leq \max\left\{\left\lceil\frac{n+d-2}{2}\right\rceil,\left\lceil 2d-\frac{d(d+1)}{n}\right\rceil\right\}.$$

The paper states that this conjecture is verified in the special cases $d\in\{2,3\}$; the general case is not asserted to be solved here.