K6 and K6,6 sequence rank formula for four-dimensional rigidity

Let $K_n$ be the complete graph and let $F\subseteq E(K_n)$. A proper $\{K_6,K_{6,6}\}$-sequence is the sequence construction defined in the source using copies of $K_6$ and $K_{6,6}$; write $C_{\leq t}$ for the union of its edge sets. Four-dimensional rank conjecture. The rank function of ${\cal R}_{4,n}$ is

$$r(F)=\min\{|F\cup C_{\leq t}|-t:(C_1,\ldots,C_t)\text{ is a proper }\{K_6,K_{6,6}\}\text{-sequence in }K_n\}$$

for every $F\subseteq E(K_n)$. The conjecture is motivated by the failure of unique maximality for ${\cal R}_{4,n}$, and the source gives no resolution.