Nonemptiness conjecture for the nonzero solution set $V(\mathbf{r})^{\times}$

Let $n,k\in\mathbb{N}$ with $n\geq k$, and let $\mathbf{r}\in\Gamma(n,k)$. Let $V(\mathbf{r})^{\times}$ denote the subset of $V(\mathbf{r})$ consisting of the relevant nonzero solutions. Nonemptiness conjecture. The set $V(\mathbf{r})^{\times}$ is non-empty. This conjecture is motivated by computational results using Maple; the paper gives no proof or resolution, so the general nonemptiness question remains open.