Bukh's algebraically independent distance-avoidance conjecture

Let $A\subset\mathbb{R}_{>0}$ be algebraically independent, meaning that no nonzero polynomial with rational coefficients vanishes on any finite tuple of distinct elements of $A$. A finite colouring of $\mathbb{R}^2$ is said to avoid $A$ if no monochromatic pair of points has distance contained in $A$. Bukh's conjecture. There is a finite colouring of $\mathbb{R}^2$ containing no monochromatic pair of points whose distance is contained in $A$. The measurable analogue is known to be impossible when $A$ is unbounded, by the result of Furstenberg, Katznelson, and Weiss; whether measurability is necessary is open, including this algebraically independent case.