Prolate quadrature conjecture on the relative accuracy of coefficient coordinates

Let $\beta^{(n)}=(\beta^{(n)}_0,\beta^{(n)}_1,\dots,\beta^{(n)}_N)$ be the coefficient vector and let $\hat{\beta}^{(n)}$ be its numerical approximation computed in Step 2. Let $\varepsilon$ denote the machine accuracy, for example $\varepsilon\approx 10^{-16}$ in double precision, and let $c$ be the band limit.

Relative-accuracy conjecture. The coordinates of $\beta^{(n)}$ satisfy, for every $1\leq k\leq N$,

$$\left|\frac{\beta^{(n)}_k-\hat{\beta}^{(n)}_k}{\beta^{(n)}_k}\right|\leq\varepsilon\cdot\log(\sqrt{c}).$$

The conjecture asserts that even coordinates that are small in absolute value are computed with high relative accuracy. The source reports preliminary analysis and extensive numerical experiments supporting it, while stating that the proof remains a subject of ongoing research.