Conjecture on active complementarity in the Beta relaxation

Let $\text{\sc Beta}$ denote the convex relaxation of the ball-constrained nonconvex quadratic program defined in the paper, with matrix variable $W$ and vectors $\ell_1,\ell_2$ as above. The relaxation includes the constraint $\ell_1^T W\ell_2\geq 0$. Active-complementarity conjecture. There exists an optimal solution $W^*$ of $\text{\sc Beta}$ with

$$\ell_1^T W^*\ell_2=0.$$

The conjecture is motivated by extensive computational experiments in which the constraint was active at optimality for every tested instance. It would explain why the strengthened relaxation with $\ell_1^T W\ell_2=0$ continues to solve the tested instances, although no general proof is given here.