I would like to know what is $k^*$ the maximum $k$ such that:

$$ \exists z_1, \ldots, z_k \in\left(\mathbb{R}^+\right)^d \quad \forall i \leqslant k \quad \exists \hat{z}_i \in \mathbb{R}^d \quad \hat{z}_i \cdot z_i>0 \quad and \quad \forall j \neq i \quad \hat{z}_i \cdot z_j<0 $$

($\cdot$ is the dot product here.)

Obviously $k^* \geq d$, but I have not been able to (dis)prove that $k^* = d$. I have tried to prove it by an induction inspired from https://mathoverflow.net/a/31442/504480, but it hasn't lead me anywhere.