Free-elevation rank formula for K_{d+2}-matroids

Let $M_0$ be the rank ${{d+2}\choose{2}}$ truncation of ${\cal R}_{d,n}$, equivalently of any $K_{d+2}$-matroid on $E(K_n)$, with $n\geq d+2$. A $K_{d+2}$-sequence is a sequence of copies of $K_{d+2}$ satisfying the properness condition used in the source, and write $C_{\leq t}$ for the union of its edge sets. Free-elevation rank conjecture. The rank function of the free elevation of $M_0$ is

$$r(F)=\min\{|F\cup C_{\leq t}|-t:(C_1,\ldots,C_t)\text{ is a proper }K_{d+2}\text{-sequence in }K_n\}$$

for every $F\subseteq E(K_n)$. This would imply that the free elevation is the unique maximal $K_{d+2}$-matroid; the conjecture is stated to hold for $d=1,2,3$ and remains unresolved in general.