A question about the vector space representation of $\mathfrak{sl}_3$

Question

Let $\mathfrak{g}$ be a simplex complex Lie algebra. Looking at this question we see that the weights of the non-zero weight vectors of an irreducible $\frak{g}$-module $V$ have weights strictly less than that of the highest weight vector of $V$. However this does not seem to work for the vector space representation of $\frak{sl}_3$. It is a three-dimensional module with highest weight $\pi_1$ and lowest weight $-\pi_2$ and the remaining weight $\pi_2 - \pi_1$. With respect to the partial order on weights we do not have that $\pi_2 - \pi_1$ is less than $\pi_1$ since $$ \pi_1 - (\pi_2 - \pi_1) = 2\pi_1 - \pi_2. $$ What am I doing wrong here?