Saturation number of $T_3^{\le 2}$

Let $T_3^{\le 2}$ denote the specified binary matrix family, and let $\operatorname{sat}(n,F)$ be the minimum number of columns in an $n$-row $F$-saturated matrix. Set $F=T_3^{\le 2}$. The saturation conjecture.

$$\operatorname{sat}(n,F)=10\quad\text{for every }n\ge 7.$$

The paper constructs $F$-saturated matrices with ten columns for all $n\ge 7$, giving the upper bound. The equality is stated as a conjecture, so the corresponding lower bound remains open.