Cofactor matroid maximality and the valuation rank formula

For $n$ vertices, let $K_n$ be the complete graph, let $\mathcal C^{d-2}_{d-1}(K_n)$ be the cofactor matroid, and let $\operatorname{val}_{K_{d+2}}$ be the valuation associated with $K_{d+2}$-sequences. Cofactor matroid conjecture. For all $d\geq1$, the cofactor matroid $\mathcal C^{d-2}_{d-1}(K_n)$ is the unique maximal $K_{d+2}$-matroid on $K_n$, and $\operatorname{val}_{K_{d+2}}$ is its rank function. The case $d=3$ was recently verified in the cited joint work; the assertion for all $d$ is open.