The Ramsey-type rank-pattern dichotomy for simplices

Let $\Delta^n=\{(x_1,\ldots,x_n)\in[0,1]^n: x_1\leq\cdots\leq x_n\}$, and let $F\subseteq\Delta^n$. Write $F^c=\Delta^n\setminus F$. The rank of a set is defined by the preceding rank condition, and an $n$-pattern is a partition pattern of a finite set whose blocks each have size $n$. An $n$-pattern is essential for $F$ if it has a copy in $F\cap D^n_{a,b}$ for every $0\leq a<b\leq1$, where $D^n_{a,b}=\{x\in\Delta^n: x_i\in(a,b)\text{ for all }i\leq n\}$. Ramsey-type rank-pattern dichotomy. At least one of the following holds:

$$\operatorname{rank}(F^c)=\infty,$$

or every finite $n$-pattern is essential for $F$. This asserts a dichotomy between infinite rank of the complement and the presence of every finite pattern at every scale. The result is presented as a conjectural Ramsey-type phenomenon in the supplied text; its resolution is not specified.