High-connectivity conjecture for completability and hyperconnectivity

Let $d$ be a positive integer, and let $G$ be a graph on $n$ vertices. Write ${\mathcal{S}}_d(G)$ and ${\mathcal{H}}_d(G)$ for the completability and hyperconnectivity matroids in dimension $d$, respectively. High-connectivity conjecture. For every positive integer $d$, there exists a positive integer $k_d$ such that every $k_d$-connected graph $G$ on $n$ vertices satisfies

$$\operatorname{rank} {\mathcal{S}}_d(G) \geq dn-d^2$$

and

$$\operatorname{rank} {\mathcal{H}}_d(G) \geq dn-d^2.$$

This would extend the paper's main theorem from bipartite rigidity to completability and hyperconnectivity for highly connected graphs. The supplied text gives no resolution, and the conjecture remains open.