Prove or disprove: Let $E,F \in \mathcal{I}$. If $\text{row}(E) \subseteq \text{row}(F)$ and $\text{col}(E) \subseteq \text{col}(F)$ then $EF=FE=E$

Question

In the statement above, $\mathcal{I} \ $ is the set of idempotent binary relations on a finite set, $\text{row}(X)$ denotes the row space of the relation $X$, $\text{col}(X)$ denotes the column space of $X$, and $XY$ denotes composition of the relations $X$ and $Y$.

I wrote a code in Mathematica that indicates the statement is true for binary relations on sets of size less than 5.

I would like to see a counter example or a proof.