The necessary-and-sufficient condition conjecture for non-strict local minimizers of homogeneous QCQPs

Let

be the homogeneous quadratically constrained quadratic optimization problem considered in the paper, and let a feasible point be a non-strict local non-global minimizer. The paper conjectures that the following necessary optimality condition is sufficient: (i) there exist $\alpha\in\mathbb{R}$ and $\beta>0$ such that $-$ and

hold; and (ii) there exists a nonzero vector $\bar v$ satisfying $-$. This conjecture seeks a complete characterization of non-strict local non-global minimizers; the supplied text gives motivation from an example with infinitely many such minimizers, but does not establish sufficiency.