Woven completeness versus woven minimality for frames

Determine whether woven completeness implies woven minimality for pairs of frame systems of weighted exponentials.

Background

For frame systems of weighted exponentials, the paper proves that woven $\ell^2$-minimality is equivalent to woven completeness. However, ordinary woven minimality is stronger than woven $\ell^2$-minimality, and the equivalence established in the paper therefore does not settle whether woven completeness yields ordinary woven minimality.

The authors explicitly identify this converse implication as unknown even in the setting where both individual weighted-exponential systems are frames.