Roots-of-unity decomposition conjecture for generalized Vandermonde matrices

Let $X=\{x_0=1,x_1,\ldots,x_k\}$ be a set of pairwise distinct roots of unity, and let $B=\{b_0=0,b_1,\ldots,b_m\}$ be a set of coprime integer numbers. Let $M=(x_i^{b_j})$ be degenerate. Roots-of-unity decomposition conjecture. Then $B$ can be split as

$$B=B_1\sqcup\cdots\sqcup B_k,$$

such that all the $x_i$ are roots of unity of degree

$$\gcd\{b-b'\mid b,b'\in B_i,\ i=1,\ldots,k\}.$$

The statement concerns the structure of degeneracy loci of generalized Vandermonde matrices. The paper says that it proves this conjecture for $k\leqslant2$, while the arbitrary-$k$ formulation remains open.