Woven minimality versus woven completeness

Determine whether woven minimality always implies woven completeness for systems of weighted exponentials, without restricting the individual weighted-exponential systems to be frames.

Background

The paper distinguishes woven minimality from woven $\ell^2$-minimality. For pairs of weighted-exponential systems that are frames, the authors prove that woven $\ell^2$-minimality is equivalent to woven completeness and consequently that woven minimality implies woven completeness under the frame assumption.

The unresolved issue is whether the implication holds in full generality, when the two individual weighted-exponential systems are not assumed to be frames. The subsequent corollary establishes the implication only under the frame hypothesis, so it does not resolve the unrestricted question.