Given a hyperplane $\alpha^T x = \beta$ in $\mathbb R^n$, with $\beta > 0, \alpha_i > 0$ for all $i \in [n]$. Then for any $\{v^i\} \subseteq \{x \in \mathbb Z^n_+ \mid \alpha^T x = \beta\}$, it's obvious to see that there must have: $\{v_i\}$ forms an antichain with respect to the component-wise order. My question is, for a given set of less than $n$ positive integer vectors, to guarantee the existence of a hyperplane $\alpha^T x = \beta$ containing all of these integer points with $\alpha_ i > 0$ for all $i \in [n]$, is the antichain condition also sufficient?

Formally speaking:

Given an antichain $\{v^i\}_{i \in[d]} \subseteq \mathbb Z^n_+$ with $d< n$. (Here antichain is with respect to the component-wise order: for any $i \neq j \in [d],$ there exists $t_1, t_2 \in [n],$ such that $ v^i_{t_1}>v^j_{t_1}, v^i_{t_2}<v^j_{t_2}$.) Then: there exists a hyperplane $\alpha^T x = \beta$ containing all these integer points, and $\alpha_i > 0$ for any $i \in [n]$.