The finite-union conjecture for bounded below approximate Schauder frames

Let $\mathcal{X}$ be a separable Banach space, and let $(\{f_n\}_n,\{\tau_n\}_n)$ be an approximate Schauder frame (ASF) for $\mathcal{X}$, meaning that its frame operator

$$S_{f,\tau}x=\sum_{n=1}^{\infty}f_n(x)\tau_n$$

is a well-defined bounded linear invertible operator. The ASF is bounded below when

$$\inf_{n\in\mathbb{N}}|f_n(\tau_n)|>0.$$

An approximate Riesz sequence (ARS) is an ASF for which there is a finite partition $Q_1,\dots,Q_N$ of $\mathbb{N}$ such that, for every $1\leq j\leq N$,

$$\inf_{n\in Q_j}\left(|f_n(\tau_n)|-\sum_{m\in Q_j,\ m\neq n}|f_n(\tau_m)|\right)>0.$$

Finite-union conjecture. Every bounded below ASF can be partitioned as a finite union of ARSs. The source's statement uses “ARBs,” but the preceding definition introduces ARSs; this row follows the defined term. The conjecture concerns extending the finite-union phenomenon from Hilbert-space frames to approximate Schauder frames on separable Banach spaces, and the source provides no resolution.