The $R_\epsilon$-conjecture for unit norm Riesz basic sequences

Let $\epsilon>0$. A unit norm Riesz basic sequence is a Riesz basic sequence ${f_i}$ whose vectors satisfy $|f_i|=1$; an $\epsilon$-Riesz basic sequence has Riesz bounds $1-\epsilon$ and $1+\epsilon$.

$R_\epsilon$-conjecture. Every unit norm Riesz basic sequence is a finite union of $\epsilon$-Riesz basic sequences.

This conjecture was posed by Casazza and Vershynin, and the source says it is equivalent to the Kadison–Singer problem. The paper also states that its finite-dimensional quantitative version is needed for the subsequent argument.