Valuation formulas for maximal hyperconnectivity, completion and rigidity matroids

Let $K_n$ be the complete graph. The $d$-hyperconnectivity matroid $\mathcal H_n^d$ is a $\{K_{d+2},K_{d+1,d+1}\}$-matroid, $\mathcal I_n^2$ is a $\{K_5,K_{3,3}\}$-matroid, and $\mathcal R_n^d$ is a $\{K_{d+2},K_{d+2,d+2}\}$-matroid. Valuation maximality conjecture. (a) For $n\geq2d+2$, $\mathcal H_n^d$ is the unique maximal $\{K_{d+2},K_{d+1,d+1}\}$-matroid on $K_n$, and its rank function is $\operatorname{val}_{\{K_{d+2},K_{d+1,d+1}\}}$. (b) For $d=2$ and $n\geq6$, $\mathcal I_n^2$ is the unique maximal $\{K_5,K_{3,3}\}$-matroid on $K_n$, and its rank function is $\operatorname{val}_{\{K_5,K_{3,3}\}}$. (c) For $d\geq3$ and $n\geq2d+4$, $\mathcal R_n^d$ is the unique maximal $\{K_{d+2},K_{d+2,d+2}\}$-matroid on $K_n$, and its rank function is $\operatorname{val}_{\{K_{d+2},K_{d+2,d+2}\}}$. These are proposed strengthenings of the preceding maximality conjecture; no resolution is given in the supplied text.