Non-negative least-squares column-exclusion conjecture

Let $G$ be a matrix with all entries non-negative and all columns linearly independent. For a column ${\bf g}_*$ of $G$, let $G_*$ be the matrix obtained by excluding ${\bf g}_*$ from $G$, and define

$$\hat{\bf x}:=\arg\min_{\bf x}\|{\bf g}_*-G_*{\bf x}\|_2^2.$$

Non-negative least-squares column-exclusion conjecture. There exists at least one column ${\bf g}_*$ of $G$ such that $\hat{\bf x}$ has non-negative entries.

The conjecture is introduced to support the effectiveness of the paper's backtracking algorithm for selecting columns in a non-negative least-squares problem. Its resolution is not specified in the supplied text.