Discrete covering-system convex-hull periodicity conjecture

Let $f$ be a mapping from $\{1,2,\ldots,n\}$ to subsets of $\{1,2,\ldots,n\}$ such that

$$\bigcup_{i=1}^n f(i)=\{1,2,\ldots,n\}.$$

For a finite set $A\subseteq\mathbb{N}$, define

$$\operatorname{conv}f(A)=\{\min f(A),\ldots,\max f(A)\},$$

the convex hull of $f(A)$. Let $I_1,\ldots,I_k$ be any partition of $\{1,2,\ldots,n\}$ into consecutive blocks,

$$I_1=\{1,2,\ldots,i_1\},\quad I_2=\{i_1+1,\ldots,i_2\},\quad \ldots,\quad I_k=\{i_{k-1}+1,\ldots,n\}.$$

Discrete convex-hull conjecture. There exist $j\in\{1,2,\ldots,k\}$ and $r,s\in I_j$ such that, for some $l\leq k$,

$$(\operatorname{conv}f)^l(\{r,s\})\supseteq\{r,s\}.$$

The paper proposes this conjecture as a discrete statement implying the covering-system conjecture; its general status is not resolved in the supplied text.