Balazs's weighted lower semi-frame conjecture for complete families

Let $\mathcal{F}$ be a family in a separable Hilbert space $\mathcal{H}$. The family is complete if its closed linear span is $\mathcal{H}$, and it is a weighted lower semi frame if some positive weighting of its elements is a lower semi frame for $\mathcal{H}$. Balazs's conjecture. If $\mathcal{F}$ is complete, then $\mathcal{F}$ is a weighted lower semi frame for $\mathcal{H}$. Theorem 3.1 proves this for families containing a complete and minimal subfamily, so the conjecture remains open for arbitrary complete families.