The quantitative $R_\epsilon$-conjecture for finite Riesz basic sequences

Let $\epsilon>0$ and $A,B>0$. A unit norm Riesz basic sequence ${f_i}_{i=1}^n$ for $\ell_2^n$ has Riesz basis bounds $0<A\leq B$ when

$$A^2\sum_i|a_i|^2\leq\left\|\sum_i a_i f_i\right\|^2\leq B^2\sum_i|a_i|^2.$$

Quantitative $R_\epsilon$-conjecture. There is a natural number $r=r(\epsilon,A,B)$ such that, for every $n\in\mathbb N$ and every such sequence, there is a partition ${A_j}_{j=1}^r$ of ${1,\mathinner{\cdotp\cdotp\cdotp},n}$ for which each ${f_i}_{i\in A_j}$ is an $\epsilon$-Riesz basic sequence.

The paper introduces this finite-dimensional quantitative form because it is the version required to connect paving for triangular operators with the Riesz-sequence conjecture. The source does not state an independent resolution status for this quantitative formulation.